# Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Save this PDF as:
Size: px
Start display at page:

Download "Chapter 4 Contravariance, Covariance, and Spacetime Diagrams"

## Transcription

1 Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz Trnsformtion, it is necessry to drw coordinte systems tht re skewed to ech other rther thn to use the trditionl orthogonl coordinte systems. It is therefore pproprite to digress for moment nd look into some of the chrcteristics of skewed coordinte system. For simplicity nd clrity we will strt our discussion in two dimensions, lter we will extend the discussion to more thn two dimensions. Consider the skewed coordinte system shown in figure 4.. We will use the stndrd nottion tht is used in reltivity nd use superscripts to lbel the coordintes x l nd x s shown. (x does not men x squred, it is just different mens of lbeling the coordintes, the reson for which, will become cler in moment.) A series of lines everywhere prllel to these coordinte xis estblishes spce grid. The intersection of ny of these two lines estblishes set of coordintes for ny prticulr point considered. Let us now drw the vector r in this coordinte system. Now let us find the components of the vector r in this skewed coordinte system. But how do we find the components of vector in skewed coordinte system? r x x Figure 4. A skewed coordinte system. () Rectngulr Components of Vector in n Orthogonl Coordinte System. First, let us recll how we find the components of vector in n orthogonl coordinte system. To find the x-component of the vector we project r onto the x-xis by dropping perpendiculr line from the tip of r down to the x-xis s shown in figure 4.(). Its intersection with the x-xis, we cll the x-component of the vector. Note tht the line perpendiculr to the x-xis y y y r r j y j i x x () (b) Figure 4. An orthogonl coordinte system. i x x 4-

2 is lso prllel to the y-xis. The y-component is found by projecting r onto the y-xis by dropping perpendiculr line from the tip of r to the y-xis. Its intersection is clled the y-component of the vector r. Also note tht the line perpendiculr to the y-xis is prllel to the x-xis. In terms of the unit vectors i nd j, nd the x nd y-components, the vector r cn be expressed s r ix + jy (4.) The set of vectors i nd j re sometimes clled set of bse vectors. Implied in the representtion of the vector r by eqution 4. is the prllelogrm lw of vector ddition, becuse ix is vector in the x-direction nd jy is vector in the y-direction. Moving these vectors prllel to themselves genertes the prllelogrm, nd the digonl of tht prllelogrm represents the sum of the two vectors ix nd jy s shown in figure 4.(b). Exmple 4. Rectngulr components of vector. A vector r hs mgnitude of 5 units nd mkes n ngle of with the x-xis. Find the x- nd y-components of this vector. Solution The x-component of the vector r is found s x r cos θ x 5 cos x 4.33 y y r The y-component of the vector r, is found s y r sin θ y 5 sin y.50 θ x x Figure 4.3 The rectngulr components of vector. To go to this Interctive Exmple click on this sentence. (b) Contrvrint components of vector. Now let us return to the sme vector in the skewed coordinte system. We introduce new system of bse vectors l nd s shown in figure 4.4. The bse vector l is in the direction of the x l xis nd is in the direction of the x xis. The bse vectors nd will be clled unitry vectors lthough they don t necessrily hve to be unit vectors. We return to the originl question. How do we find the components of r? For the orthogonl system, the perpendiculr from the tip of r ws perpendiculr to one xis nd prllel to the other. For the skewed coordinte system the prllel of one xis is not perpendiculr to the other. So there ppers to be two wys to find the components of vector in skewed coordinte system. For the first component let us drop line from the tip of r, prllel to the x xis, to the x l xis. This component will be clled the contrvrint component of the vector r nd will be designted s x nd is shown in red in figure 4.4(). Drop nother line, prllel to the x xis, to the x xis. This gives the second contrvrint component x, which is lso shown in red in figure 4.4(). In terms of these contrvrint components the vector r cn be written s 4-

3 x x x x cos α α x r α r α x θ α α x x cos α x x α O x () Contrvrint components (b) Prllelogrm lw Figure 4.4. Contrvrint components of vector. x r x + x (4.) We observe from figure 4.4(b) tht the vectors x nd x dd up to the vector r by the prllelogrm lw of vector ddition. So tht eqution 4. is vlid representtion of vector in the skewed coordinte system. Exmple 4. Contrvrint components of vector. A vector r hs mgnitude of 5.00 units nd mkes n ngle of with the x-xis. If the skewed coordinte system, figure 4.5, mkes n ngle α , () find the contrvrint components of this vector, nd (b) express the vector in terms of its contrvrint components. Solution. The contrvrint components of the vector r re found from the geometry of figure 4.5. The contrvrint component x is found by observing from tringle I sin α r sin (α θ) (4.3) x Upon solving for the contrvrint component x we get r sin ( α - θ) x (4.4) x 5.00 sin( ) 5.00 (0.643) 5(0.684) sin x 3.4 x (α θ) α θ α α x x x x x Figure 4.5 Contrvrint components. r I x r sin (α θ) x II r sin θ The contrvrint component x is found by observing from tringle II tht 4-3

4 sin α r sin θ x (4.5) nd upon solving for the contrvrint component x we get r sin θ x (4.6) x 5.00 sin (0.500) sin x.66 5(0.53) b. The vector r cn now be written in terms of its contrvrint components from eqution 4. s r As check tht these re the correct contrvrint components of the vector r, let us determine the mgnitude r from this result. We cn no longer use the Pythgoren Theorem to determine r, becuse we no longer hve right tringle s we do in the cse of rectngulr components. We cn however pply the lw of cosines to the tringle of figure 4.4(b) to obtin r (x ) + (x ) + x x cos α (3.4) + (.66) + (3.4)(.66) cos( ) (.7) + (7.08) + (8.) (0.34) r 5.00 We see tht we do get the correct result for the mgnitude of the vector r. To go to this Interctive Exmple click on this sentence. (c) Covrint components of vector. For second representtion of the components of vector in skewed coordinte system, we now drop perpendiculr from the tip of r to the x xis intersecting it t the point tht we will now designte s x, nd is shown in yellow in figure 4.6(). We will cll x covrint component of the vector r. Now drop perpendiculr from the tip of r to the x xis, obtining the second covrint component x, lso shown s yellow line. We now hve the two vector components x nd x. But these vector components do not stisfy the prllelogrm lw of vector ddition when we try to dd them together hed to til, s is obvious from figure 4.6(b). Tht is, by dding the vectors from hed to til, you cn see tht x + x will not dd up to the vector r. In fct you cn see tht the sum would be ctully greter thn the mgnitude of r, nd would not be in the correct direction. Therefore in terms of these components r x + x At first glnce it therefore seems tht the only wy we cn find the components of vector tht is consistent with the prllelogrm lw of vector ddition is to drop lines from the tip of r tht re prllel to the coordinte xis, thereby obtining the contrvrint components of vector. 4-4

5 x x r α θ () x x (b) Figure 4.6 Covrint components of vector. However there is still nother wy to determine the components of the vector r nd tht is to estblish new coordinte system with unit vectors e nd e where e is perpendiculr to nd e is perpendiculr to. This new bse system is shown in figure 4.7(), long with the old bse system. The bse vector e defines the direction of new xis x, while the bse vector e defines L L L L () (b) Figure 4.7 Introduction of some new bse vectors. 4-5

6 new xis x. We now drop perpendiculr from the tip of r to the x xis, but insted of terminting the perpendiculr t the x xis, we continue it down until it intersects the new x xis. (Note tht the perpendiculr line is perpendiculr to the x -xis but not the x xis.) We cll the projection on the new xis, L. Now drop perpendiculr from the tip of r to x, then extrpolte it until it crosses the new x xis t the point L. Then we cn see from figure 4.7, tht the vectors Le nd Le will dd up by the prllelogrm lw of vector ddition to r Le + Le (4.7) Cn we express the lengths L nd L, in terms of the covrint components x nd x? Referring bck to figure 4.4(), we first note tht the ngle α is the mesure ngle of the mount of skewness of the coordintes. The covrint component x cn be seen to be composed of two lengths, i.e. x x + x cosα (4.8) while the covrint component x is composed of the two lengths From the bottom tringle in figure 4.7() we observe tht x x + x cosα (4.9) nd hence L x x L (4.0) And from the upper tringle in figure 4.7() we hve Therefore L x x L (4.) Replcing equtions 4.0 nd 4. into eqution 4.7 gives r x e + x e (4.) or upon slightly rerrnging terms, we cn write this s r x e + x e If we now define the two new bse vectors nd e sin α e sin α (4.3) (4.4) 4-6

7 then the vector r cn be written in terms of the covrint components s r x + x (4.5) Since eqution 4.5 is just eqution 4.7, but in different nottion, it lso stisfies the prllelogrm lw of vector ddition. Exmple 4.3 Covrint components of vector. A vector r hs mgnitude of 5.00 units nd mkes n ngle of with the x-xis. If the skewed coordinte system mkes n ngle α , () find the covrint components of this vector, (b) express the vector in terms of its covrint components, nd (c) find the vlues of the bse vectors. Solution. To express the vector in terms of its covrint components we use eqution 4.5 r x + x The covrint component x is found from figure 4.7() s while the covrint component x is found from figure 4.7() s x r cos θ (4.6) x 5.00 cos ( ) 5.00 (0.866) x 4.33 x r cos (α θ) (4.7) x 5.00 cos ( ) 5.00 cos ( ) x 3.83 As check let us find the covrint component x from eqution 4.8 in terms of the contrvrint components s x x + x cosα (4.8) The vlues of x nd x were determined in Exmple 4. to be nd x r sin (α θ) (4.4) sin α x 5.00 sin( ) 3.4 sin 70 0 x r sin θ sin α x 5.00 sin sin 70 0 (4.6) Replcing these vlues into eqution 4.8 gives for the covrint component x x x + x cosα (4.8) x cos

8 x 4.33 The covrint component x is found from eqution 4.9 in terms of the contrvrint components s x x + x cosα (4.9) x cos x 3.83 Notice tht the components re the sme for either procedure. b. The vector r in terms of the covrint components is obtined from eqution 4.5 s Hence, r x + x r As check tht these re the correct covrint components of the vector r, let us determine the mgnitude r from this result. We cn no longer use the Pythgoren Theorem to determine r, becuse we no longer hve right tringle s we do in the cse of rectngulr components. We cn however pply the lw of cosines to the tringle of figure 4.4(b) to obtin r (x) + (x) xx cos α r (4.33) + (3.83) (4.33)(3.83) cos 70 r 4.70 units But something is wrong here. We know the mgnitude of r should be 5 nd it is not. The trouble is tht the unitry vectors re not unit vectors but unitry vectors. They re not equl to one. We hve to tke these vectors into ccount. They re tken into ccount by using the bse vectors e nd e which re unit vectors. c. The bse vector is found from eqution 4.3 s e e e sin e (4.3) The bse vector is found from eqution 4.4 s e e e sin e (4.4) Notice tht the bse vectors nd re not unit vectors, nd their vlue will vry depending upon the ngle α tht the coordintes re skewed. If we wish we could lso write the vector in terms of the unit vectors e nd e by using equtions 4.3 nd 4.4. r x + x r r (4.33)(.06)e + (3.83)(.06)e 4-8

9 r (4.6)e + (4.08)e Recll from eqution 4.7 nd figure 4.7b tht r Le + Le (4.7) where L x times the mgnitude of the unitry vector,. Tht is nd L x L (4.33)(.06) 4.6 L (3.83)(.06) 4.08 As check tht these re the correct covrint components of the vector r, let us determine the mgnitude r from this result. As we showed erlier for the contrvrint vector, we cn no longer use the Pythgoren Theorem to determine r, becuse we no longer hve right tringle s we do in the cse of rectngulr components. We cn however pply the lw of cosines to the tringle of figure 4.7(b) to obtin r (L) + (L) LL cos α r (4.6) + (4.08) (4.6)(4.08) cos 70 r 5.0 r 5.00 units Notice tht we get the sme mgnitude of 5 units s we did in exmples 4. nd 4.. We could lso use eqution 4.7 by first determining the vlues of L nd L from the equtions nd L x / sin α L (4.33) / sin (70) L (4.6) L x / sin α L (3.83) / sin (70) L (4.08) nd plcing these into the lw of cosines we get r (L) + (L) LL cos α r (4.6) + (4.08) (4.6)(4.08) cos 70 r 5.0 r 5.00 units Agin notice tht we get the sme correct result for the mgnitude of the vector r. To go to this Interctive Exmple click on this sentence. To summrize our results, eqution 4. is the representtion of the vector r in terms of its contrvrint components, r x + x (4.) while eqution 4.5 is the representtion of the vector r in terms of its covrint components, i.e. 4-9

10 r x + x (4.5) So in skewed coordinte system there re two types of components- contrvrint nd covrint. The contrvrint components re found by prllel projections onto the coordinte xes while the covrint components re found by perpendiculr projections. Contrvrint components re designted by superscripts, x i, while covrint components re designted by subscripts, xi. The bse vectors nd re not unit vectors even if nd re. The distinction between contrvrint nd covrint components disppers in orthogonl coordintes, becuse the xes re orthogonl. Tht is, in orthogonl coordintes, projection which is prllel to one xis, is lso perpendiculr to the other. Let us now return to the spcetime digrms we discussed in chpter nd see how these concepts of covrince nd contrvrince re pplied to these spcetime digrms. 4. Different Forms of The Spcetime Digrms Figure.9 showed the reltion between the S nd S frmes of reference in spcetime. The S frme ws the sttionry frme nd S ws frme moving to the right with the velocity v. The ngle θ, of figure.9 ws given by eqution.3 s θ tn But we lredy sid tht there is no frme of reference tht is bsolutely t rest, nd yet our digrm shows the preferred sttionry frme, S, s n orthogonl coordinte system while the moving frme, S, is n cute skewed coordinte system. So it seems s if the rest frme is specil frme compred to the moving frme. However, the principle of reltivity sys tht ll frmes re equivlent. Tht is, there should be no distinction between frme of reference tht is t rest or one tht is moving t constnt velocity v. Figure 3.9 should be modified to show tht there is no preferentil frme of reference. We showed in Chpter, tht if body is t rest nd body moves to the right with velocity v, tht this is equivlent to body being t rest nd body moving to the left with the velocity v. Another equivlence is to hve n rbitrry observer t rest between nd nd body cn move to the right with velocity v/ with respect to the frme t rest nd body cn move to the left with velocity v/. We cn incorporte these generlities by redrwing figure 3.9 for S with the τ xis now mking n ngle θ/ with the originl τ-xis, nd by showing second observer, S, moving to the left of the sttionry observer with the velocity v/. This is shown in the spcetime digrm of figure 4.8 s the τ xis mking n ngle θ/ with the τ-xis. The ngle θ/ is computed in the sme wy s the computtion for the τ -xis, tht is, v c θ/ tn v/ c θ/ tn v/ c S frme moving to the right with velocity v/ with respect to S frme S frme moving to the left with velocity v/ with respect to S frme In this wy the S frme will be moving t the velocity v with respect to the S frme. Similrly n x -xis cn be drwn t n ngle θ/ from the x-xis. Notice tht the x nd τ re found in the sme wy tht we found x nd τ, except tht x nd τ hve negtive slopes, indictive of the motion to the left. These new x - nd τ -xes generte new cute skewed coordinte system, S, locted in the fourth qudrnt, s seen in figure 4.8() nd 4.8(b). Note tht the S coordinte system is shown in blue while the S coordinte system is shown in red. Also notice tht becuse of the symmetry, the scles re the sme in the S frme s they re in the S frme, which is of course different to the scle in the S frme s we showed before. Also note tht in figure.9, θ ws the ngle between τ nd τ becuse S ws moving t the speed v with respect to the S frme. Now 4-0

11 notice tht θ/ is the ngle between τ nd τ becuse S is now moving t the speed v/ with respect to the S frme of reference. Also note tht θ is now the ngle between τ nd τ becuse S is now moving t the speed v with respect to S. θ/ θ/ θ/ θ/ θ θ/ θ/ θ θ/ θ/ () (b) Figure 4.8 Reltion of S nd S frme of references. Also note from figure 4.8 tht the x -xis is orthogonl to the τ -xis since but α 90 0 θ. Hence, nd τ 0x α + θ τ 0x 90 0 θ + θ τ 0x 90 0 Similrly, the τ -xis is perpendiculr to the x -xis, since but φ 90 0 θ/. Hence, nd x 0τ φ + θ/ x 0τ 90 0 θ/ + θ/ x 0τ 90 0 The fct tht the x -xis is orthogonl to the τ -xis, nd the τ -xis is orthogonl to the x -xis, should remind us of how the x xis ws perpendiculr to the x xis nd the x xis ws perpendiculr to the x xis in figure 4.7 in our study of some of the chrcteristics of covrint nd contrvrint components. We will return to this similrity shortly. Figure 4.8 shows tht the S nd S frmes re symmetricl with respect to the S frme of reference, but not with respect to ech other. Both frmes should lso mesure the sme velocity of light c, which is ssured if the light line OL were to bisect both sets of coordinte xes. Also note tht becuse of the symmetry of both S nd S frmes, they would both intersect the scle hyperbols t the sme vlues. Hence, the scle on the S frme is the sme s the scle on the S frme. We cn modify figure 4.8 by reflecting the x -xis in the fourth qudrnt, through the origin of the coordintes, to mke n x -xis in the second qudrnt, s shown in figure 4.9. Note tht now 4-

12 the light line OL does indeed bisect the x,τ -xes nd the x,τ -xes gurnteeing tht the speed of light is sme in both coordinte systems. The S coordinte system is now n obtuse skewed coordinte system insted of the cute one it ws in the fourth qudrnt. Figure 4.9 should now be used to describe events in the S nd S coordinte systems, insted of figure 3.9 which described events in the S nd S frmes of reference. θ/ θ/ θ θ/ θ/ θ/ θ θ/ θ/ θ θ/ () (b) Figure 4.9 New S nd S frme of references. 4.3 Reciprocl Systems of Vectors We hve discussed the spcetime digrms in two dimensions. We would like to extend tht discussion first into three dimensions nd then into four or more dimensions. In order to extend this discussion we must first discuss the concept of reciprocl systems of vectors. Consider the three dimensionl oblique coordinte system shown in figure 4.3. The three xes re described by the constnt unitry vectors,, nd 3 s shown in the figure. We now define the set of reciprocl unitry vectors s 3 3 (4.6) Figure 4.3 The reciprocl unitry vectors (4.7) (4.8) 4-

13 By nture of the cross product of two vectors, nd s cn be seen in figure 4.3, is perpendiculr to the plne generted by nd 3; is perpendiculr to the plne generted by 3 nd ; nd 3 is perpendiculr to the plne generted by nd. Hence,,, nd 3 re clled unitry vectors, while the vectors,, nd 3 re clled reciprocl unitry vectors. Let us now consider combintions of products of these unitry vectors nd their reciprocl unitry vectors. First, let us consider the product nd 3 3 ( 3) ( 3) 3 (4.9) (4.0) But s you recll from vector nlysis, by the vector triple product of three vectors, cyclic interchnge of letters is permissible, tht is, Applying this to our unitry vectors we get Using eqution 4. in eqution 4.0 gives ( b c) b ( c ) c ( b ) (4.) ( 3) 3 3 (4.) Agin using eqution 4.8 we find for the third product ( 3) 3 3 (4.3) The results of equtions 4.9, 4., nd 4.3 shows tht the product of unitry vector nd its reciprocl unitry vector is equl to one, tht is, (4.4) 3 3 When we consider the mixed products of these unitry vectors nd their reciprocl unitry vectors we get 3 ( 3 ) ( 3) ( 3) But s cn be seen in figure 4.3, 3 is perpendiculr to the plne generted by nd 3 nd hence is perpendiculr to the vector nd hence its dot product with is equl to zero. Tht is, Hence (3 ) 3 cos

14 Similrly the mixed product 3 ( 3) 3 But s cn lso be seen in figure 4.3, is perpendiculr to the plne generted by nd nd hence is perpendiculr to the vector nd hence its dot product with is equl to zero. Tht is, ( ) cos Hence the dot product of ( ) is equl to zero, therefore 3 0 In similr wy, ll the mixed products of the unitry vectors nd the reciprocl unitry vectors re equl to zero. Tht is, (4.5) In summry, the reciprocl unitry vectors re defined by equtions 4.6, 4.7, nd 4.8, nd the product of these unitry vectors nd the reciprocl unitry vectors re summrized in equtions 4.4 nd 4.5. Just s equtions expressed the reciprocl unitry vectors in terms of the unitry vectors, the unitry vectors cn be expressed in terms of the reciprocl unitry vectors by the sme reciprocl reltions. Tht is, 3 (4.6) (4.7) (4.8) Since the order of dot product is not significnt, tht is, b b, the combintions of ll the products in eqution 4.9 through 4.5 re the sme. Tht is, nd get (4.9) (4.30) If we pply the sme resoning process to the orthogonl i, j, k, system of unit vectors we j k i i i i ( j k) i i k i j j j i ( j k) i i i j k k k i j k i i 4-4

15 Therefore, the reciprocl vectors of i, j, k, re the vectors i, j, k, themselves. In fct for ny orthogonl set of unit vectors, whether rectngulr, sphericl, cylindricl etc., the reciprocl unit vectors will be the unit vectors themselves. All orthogonl sets of vectors re self-reciprocl. The only time we will hve reciprocl sets of vectors is when we hve oblique coordinte systems, s we do in our spcetime digrms. In section 4. we nlyzed skewed coordinte system in two-dimensions nd showed tht we could represent vector in tht two-dimensionl system by using either contrvrint or covrint components of vector. Tht is, we found the vector r could be written in terms of the contrvrint components x nd x s r x + x (4.) nd in terms of the covrint components x nd x s r x + x (4.5) Remember tht the contrvrint components were found by dropping lines tht were prllel to the pproprite xes, while the covrint components were found by dropping lines tht were perpendiculr to the pproprite xes. Now tht we hve estblished three dimensionl skewed coordinte system, we cn now write the vector r, in figure 4.3, in terms of the three dimensionl contrvrint components x, x, nd x 3, nd the unitry vectors,, nd 3 s r x + x + x 3 3 (4.3) r Figure 4.3 Three dimensionl skewed coordinte system. The vector r cn lso be expressed in terms of the covrint components x, x, nd x3 of the vector nd the reciprocl system of unitry vectors,, nd 3 s r x + x + x3 3 (4.3) where the reciprocl unit vectors,, nd 3 re given by equtions 4.6, 4.7, nd 4.8. Exmple 4.4 In section 4. we showed tht we could estblish new coordinte system with unit vectors e nd e where e is perpendiculr to nd e is perpendiculr to. The bse vector e defined 4-5

16 the direction of new xis x, while the bse vector e defined new xis x. In this new set of coordinte we showed tht vector r could be written in terms of the covrint components s if we defined the two new bse vectors nd r x + x (4.5) e e (4.3) (4.4) Show tht eqution 4.3 is equivlent to eqution (4.6) The reciprocl unitry vector is given by Solution 3 3 But the ngle between the unitry vectors nd 3 is the skew ngle α of the coordintes. Hence, 3 3 sin α sin α Where 3 since they re unit vectors. Also since the ngle between the vectors nd ( 3) is (90 0 α), then ( 3) ( 3) cos (90 0 α) 3 sin α cos (90 0 α) But 3 since they re unit vectors. Therefore However, Therefore ( 3) sin α cos (90 0 α) cos (90 0 α) cos 90 0 cos α + sin 90 0 sin α sin α ( 3) sin α cos (90 0 α) sin α Replcing these vlues in eqution 4.6 gives nd hence sin α 3 3 e Hence the cse shown in section 4. for finding the covrint components of vector is specil cse of the reciprocl system of vectors. 4-6

17 4.4 Exmple of The Use of Covrint nd Contrvrint Vectors As we hve seen, ny vector cn be written in two wys. One in terms of the contrvrint components nd the other in terms of the covrint components. A vector written in terms of its contrvrint components is clled contrvrint vector. A vector written in terms of its covrint components is clled covrint vector. As n exmple, force vector cn be written s or F F + F Contrvrint Vector ( 4.33 ) F F + F Covrint Vector (4.34) Notice tht the contrvrint vector is represented in terms of the contrvrint components nd the bse vectors nd, while the covrint vector is represented in terms of the covrint components nd the bse vectors nd. Either bse system or both my be used in connection with the vectoril tretment of given problem. As n exmple, the work done in moving n object through displcement r by force F cn be expressed three wys: () in terms of contrvrint vectors (b) in terms of covrint vectors (c) in terms of mixture of contrvrint nd covrint vectors. () Work done using contrvrint vectors. The work done in terms of the contrvrint vectors is Now W F r (F + F ) (x + x ) (4.35) W F x + F x + F x + F x cos0 0 (4.36) cos0 0 (4.37) If the ngle between the two xis is α then Therefore the work done in terms of the contrvrint components is which is not prticulrly simple nd is dependent upon the ngle α. (b) The work done using covrint vectors. The work done in terms of the covrint vectors is Now we showed in equtions 4.3 nd 4.4 tht cos α cosα (4.38) cos α cosα (4.39) W F x + F x + (F x + F x ) cosα (4.40) W F r (F + F ) (x + x ) (4.4) W Fx + Fx + Fx + Fx (4.4) 4-7

18 Therefore, nd e nd e sin α 0 cos0 (4.43) sin α 0 cos0 (4.44) Now the ngle between nd is (80 0 α) s cn be seen in figure 4.7. Therefore but Therefore 0 cos(80 α ) cos(80 0 α) cos(80 0 ) cos α sin (80 0 ) sin( α) cos α cos α sin α cos α sin α (4.45) (4.46) Substituting equtions 4.43 through 4.45 into eqution 4.4 gives for the work done W Fx + Fx Fx cos α Fx cos α sin α sin α sin α sin α (4.47) which is rther complicted form for the work done (c) The work done using mixture of contrvrint nd covrint components. The work done cn be expressed s the product of the contrvrint force vector nd the covrint displcement vector. Tht is, W F r (F + F ) (x + x ) (4.48) W F x + Fx + F x + F x (4.49) But s cn be seen in figure 4.7, nd shown in eqution 4.30 nd by eqution becuse 0 becuse Replcing these vlues into eqution 4.49 gives W F x + F x (4.50) Eqution 4.50 gives the work done expressed in terms of contrvrint nd covrint components. If we hd expressed the force s covrint vector nd the displcement s the contrvrint vector we would hve obtined W F r (F + F ) (x + x ) 4-8

19 W F x + F x (4.5) In generl the product of contrvrint vector with covrint vector will yield n invrint (sclr) which will be independent of the coordinte system used. Hence, when using skewed coordinte systems, it is n dvntge to hve two reciprocl bse systems. In most of the nlysis done in generl reltivity by tensor nlysis, there will usully be mix of covrint nd contrvrint vectors. Also note tht the unitry vectors nd cn hve ny mgnitude. As n exmple if nd l then the spce grid would look s in figure 4.33(). If on the other hnd nd l, the spce grid would pper s in figure 4.33(b). We see tht this mounts to hving different scle on ech xis. Tht is, unit length on the x xis is twice s lrge s the unit length on the x xis. Although l nd cn hve ny mgnitude in generl, we will lmost lwys let them hve unit mgnitude () l (b) ; Equl Spced Grid Different scle on x nd x xis. Figure 4.3 The unitry vectors nd cn hve ny mgnitude. Note tht the contrvrint vector in eqution 4.33 hs the contrvrint components of the vector, F nd F, nd the unitry bse vectors nd ; while the covrint vector in eqution 4.34 is represented in terms of the covrint components of the vector F nd F nd the reciprocl unitry bse vectors nd. F F + F Contrvrint Vector (4.33 ) nd F F + F Covrint Vector (4.34) Notice tht the unitry bse vectors re described with subscripts, while the reciprocl unitry bse vectors re described with superscripts. Notice tht the product of ech term is product of contrvrint superscript nd covrint subscript. Hence, the vector cn be thought of s consisting of contrvrint nd covrint terms; superscript times subscript for contrvrint vector nd subscript times superscript for covrint vector. Since the product of contrvrint vector nd covrint vector gives us n invrint quntity or constnt, this nottion will give us n invrint quntity for the mgnitude of ny vector. We will see much more of this lter. In generl, the product of contrvrint vector with covrint vector will yield n invrint (sclr) which will be independent of the coordinte system used. Hence when using skewed coordinte systems, it is n dvntge to hve two reciprocl bse systems. We will see tht in most of the nlysis done in generl reltivity, there will usully be mix of covrint nd contrvrint vectors. Summry of Bsic Concepts Skewed Coordinte Systems. In order to show inertil motion tht is consistent with the Lorentz Trnsformtion, it is necessry to drw coordinte systems tht re skewed to 4-9

20 ech other rther thn to use the trditionl orthogonl coordinte systems. Components of vector in rectngulr coordinte system. The x-component of vector is found by dropping perpendiculr line from the tip of r down to the x-xis. Note tht the line perpendiculr to the x-xis is lso prllel to the y-xis. The y-component is found by dropping perpendiculr line from the tip of r to the y-xis. Also note tht the line perpendiculr to the y-xis is prllel to the x-xis. Contrvrint components of vector in skewed coordinte system. For the skewed coordinte system the prllel of one xis is not perpendiculr to the other. For the first component we drop line from the tip of r, prllel to the x xis, to the x l xis. This component is clled the contrvrint component of the vector r nd is designted s x. Drop nother line, prllel to the x xis, to the x xis. This gives the second contrvrint component x. The bse vector l is in the direction of the x l xis nd is in the direction of the x xis. The bse vectors nd re clled unitry vectors lthough they don t necessrily hve to be unit vectors. The vector r cn be written in terms of the contrvrint components s r x + x Covrint components of vector in skewed coordinte system. For second representtion of the components of vector in skewed coordinte system, we drop line from the tip of r, perpendiculr to the x xis intersecting it t the point tht we will now designte s x, We will cll x covrint component of the vector r. We now drop line from the tip of r perpendiculr to the x xis, obtining the second covrint component x. We now hve the two vector components x nd x. However, these vector components do not stisfy the prllelogrm lw of vector ddition when we try to dd them together hed to til. Tht is, r x + x, However, if we define two new bse vectors e e nd sin α sin α then the vector r cn be written in terms of the covrint components s r x + x In generl the product of contrvrint vector with covrint vector will yield n invrint (sclr) which will be independent of the coordinte system used. Hence, when using skewed coordinte systems, it is n dvntge to hve two reciprocl bse systems. We will see tht in most of the nlysis done in generl reltivity, there will usully be mix of covrint nd contrvrint vectors. In Summry, in skewed coordinte system there re two types of components- contrvrint nd covrint. The contrvrint components re found by prllel projections onto the coordinte xes while the covrint components re found by perpendiculr projections. Contrvrint components re designted by superscripts, x i, while covrint components re designted by subscripts, xi. The bse vectors nd re not unit vectors even if nd re. The distinction between contrvrint nd covrint components disppers in orthogonl coordintes, becuse the xes re orthogonl. Tht is, in orthogonl coordintes, projection which is prllel to one xis, is lso perpendiculr to the other. 4-0

21 Summry of Importnt Equtions The vector r written in terms of contrvrint components r x + x (4.) Contrvrint component x r sin ( α - θ) x Contrvrint component x r sin θ x The bse vectors e sin α e sin α (4.4) (4.6) (4.3) (4.4) The vector r written in terms of covrint components r x + x (4.5) Covrint component x x r cos θ (4.6) Covrint component x x r cos (α θ) (4.7) Lorentz trnsformtion for spce coordintes x '' x ' + vt' (4.3) v / c Lorentz trnsformtion for the time v t' + x' coordintes. t'' c v / c (4.35) Inverse Lorentz Trnsformtion for spce coordintes x ' x '' vt'' (4.36) v / c Inverse Lorentz Trnsformtion for time v t'' x'' coordintes t' c (4.37) v / c Length contrction formul L L0 v / c (4.46) Time diltion formul t'' t' t ' 0 v / c t '' 0 v / c (4.5) (4.56) The invrint intervl of Spcetime ( s ) ( x ) c ( t ) (4.94) nd ( s ) ( x ) c ( t ) (4.95) The invrint intervl of Spcetime in terms of differentil quntities (ds ) (dx ) c (dt ) (4.96) nd (ds ) (dx ) c (dt ) (4.97) The invrint intervl in four-dimensionl spcetime (ds) c (dt) (dx) (dy) (dz) (4.98) Reciprocl unitry vectors expressed in terms of the unitry vectors 3 (4.6) ( 3) 3 (4.7) ( 3) 3 (4.8) 3 Products of the unitry vectors nd the reciprocl unitry vectors 3 3 (4.4) (4.5) The unitry vectors expressed in terms of the reciprocl unitry vectors, 3 (4.6) (4.7) 4-

22 3 3 (4.8) The combintions of ll the products (4.9) nd (4.9) For three dimensionl skewed coordinte system, the vector r is written in terms of the three dimensionl contrvrint components x, x, nd x 3, nd the unitry vectors,, nd 3 s r x + x + x 3 3 (4.3) the vector nd the reciprocl system of unitry vectors,, nd 3 s r x + x + x3 3 (4.3) Work done using contrvrint vectors. W F r (F + F ) (x + x )(4.35) W F x + F x + (F x + F x ) cosα (4.40) Work done using covrint vectors. W F r (F + F ) (x + x ) (4.4) W Fx + Fx Fx cos α Fx cos α(4.47) sin α sin α sin α sin α The work done using mixture of contrvrint nd covrint components. W F r (F + F ) (x + x )(4.48) W F x + F x (4.50) W F x + F x (4.5) The vector r cn lso be expressed in terms of the covrint components x, x, nd x3 of. Why cn t we just use orthogonl systems in our nlysis of reltivity?. Wht is contrvrint vector? 3. Wht is covrint vector? 4. Is unitry vector the sme s unit vector? Questions for Chpter 4 Problems for Chpter 4 5. When using product of two vectors, is it better to hve two covrint vectors, two contrvrint vectors, or one of ech? 4. The Components of Vector in Skewed Coordintes. A vector r hs mgnitude of 5.0 units nd mkes n ngle of with the x- xis. Find the rectngulr components of this vector. A vector r hs mgnitude of 5.0 units nd mkes n ngle of with the x- xis. If the skewed coordinte system, mkes n ngle α , () find the contrvrint components of this vector, nd (b) express the vector in terms of its contrvrint components. 3. A vector r hs mgnitude of 5.0 units nd mkes n ngle of with the x- xis. If the skewed coordinte system mkes n ngle α , () find the covrint components of this vector, (b) express the vector in terms of its covrint components, nd (c) find the vlues of the bse vectors. To go to nother chpter, return to the tble of contents by clicking on this sentence. 4-

### Things to Memorize: A Partial List. January 27, 2017

Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved

More information

### Chapter 2. Vectors. 2.1 Vectors Scalars and Vectors

Chpter 2 Vectors 2.1 Vectors 2.1.1 Sclrs nd Vectors A vector is quntity hving both mgnitude nd direction. Emples of vector quntities re velocity, force nd position. One cn represent vector in n-dimensionl

More information

### DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS

3 DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS This chpter summrizes few properties of Cli ord Algebr nd describe its usefulness in e ecting vector rottions. 3.1 De nition of Associtive

More information

### Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.

1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples

More information

### The Regulated and Riemann Integrals

Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue

More information

### 20 MATHEMATICS POLYNOMIALS

0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

More information

### 7.2 The Definite Integral

7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where

More information

### Abstract inner product spaces

WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the

More information

### On the diagram below the displacement is represented by the directed line segment OA.

Vectors Sclrs nd Vectors A vector is quntity tht hs mgnitude nd direction. One exmple of vector is velocity. The velocity of n oject is determined y the mgnitude(speed) nd direction of trvel. Other exmples

More information

### 4 VECTORS. 4.0 Introduction. Objectives. Activity 1

4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply

More information

### Partial Derivatives. Limits. For a single variable function f (x), the limit lim

Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles

More information

### Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

More information

### Line and Surface Integrals: An Intuitive Understanding

Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of

More information

### How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4

More information

### 13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in

More information

### Is there an easy way to find examples of such triples? Why yes! Just look at an ordinary multiplication table to find them!

PUSHING PYTHAGORAS 009 Jmes Tnton A triple of integers ( bc,, ) is clled Pythgoren triple if exmple, some clssic triples re ( 3,4,5 ), ( 5,1,13 ), ( ) fond of ( 0,1,9 ) nd ( 119,10,169 ). + b = c. For

More information

### The Algebra (al-jabr) of Matrices

Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he lgebr (l-jbr) of Mtrices lgebr s brnch of mthemtics is much broder thn elementry lgebr ll of us studied in our high school dys. In sense

More information

### THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.

THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem

More information

### MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

More information

### Best Approximation in the 2-norm

Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion

More information

### Improper Integrals, and Differential Equations

Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted

More information

### 7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus

7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e

More information

### Lesson Notes: Week 40-Vectors

Lesson Notes: Week 40-Vectors Vectors nd Sclrs vector is quntity tht hs size (mgnitude) nd direction. Exmples of vectors re displcement nd velocity. sclr is quntity tht hs size but no direction. Exmples

More information

### Math 1B, lecture 4: Error bounds for numerical methods

Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the

More information

### Polynomials and Division Theory

Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the

More information

### Best Approximation. Chapter The General Case

Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given

More information

### Jim Lambers MAT 169 Fall Semester Lecture 4 Notes

Jim Lmbers MAT 169 Fll Semester 2009-10 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of

More information

### Math 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8

Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite

More information

### Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions

Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner

More information

### Math 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions

Mth 1102: Clculus I (Mth/Sci mjors) MWF 3pm, Fulton Hll 230 Homework 2 solutions Plese write netly, nd show ll work. Cution: An nswer with no work is wrong! Do the following problems from Chpter III: 6,

More information

### ( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that

Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we

More information

### 2. VECTORS AND MATRICES IN 3 DIMENSIONS

2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

More information

### Vectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3-dimensional vectors:

Vectors 1-23-2018 I ll look t vectors from n lgeric point of view nd geometric point of view. Algericlly, vector is n ordered list of (usully) rel numers. Here re some 2-dimensionl vectors: (2, 3), ( )

More information

### along the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate

L8 VECTOR EQUATIONS OF LINES HL Mth - Sntowski Vector eqution of line 1 A plne strts journey t the point (4,1) moves ech hour long the vector. ) Find the plne s coordinte fter 1 hour. b) Find the plne

More information

### ODE: Existence and Uniqueness of a Solution

Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =

More information

### Quadratic Forms. Quadratic Forms

Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte

More information

### Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

More information

### Vyacheslav Telnin. Search for New Numbers.

Vycheslv Telnin Serch for New Numbers. 1 CHAPTER I 2 I.1 Introduction. In 1984, in the first issue for tht yer of the Science nd Life mgzine, I red the rticle "Non-Stndrd Anlysis" by V. Uspensky, in which

More information

### INTRODUCTION. The three general approaches to the solution of kinetics problems are:

INTRODUCTION According to Newton s lw, prticle will ccelerte when it is subjected to unblnced forces. Kinetics is the study of the reltions between unblnced forces nd the resulting chnges in motion. The

More information

### Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite

Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite

More information

### Mapping the delta function and other Radon measures

Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support

More information

### Duality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.

Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we

More information

### HW3, Math 307. CSUF. Spring 2007.

HW, Mth 7. CSUF. Spring 7. Nsser M. Abbsi Spring 7 Compiled on November 5, 8 t 8:8m public Contents Section.6, problem Section.6, problem Section.6, problem 5 Section.6, problem 7 6 5 Section.6, problem

More information

### Chapter 3. Vector Spaces

3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce

More information

### Bernoulli Numbers Jeff Morton

Bernoulli Numbers Jeff Morton. We re interested in the opertor e t k d k t k, which is to sy k tk. Applying this to some function f E to get e t f d k k tk d k f f + d k k tk dk f, we note tht since f

More information

### CHAPTER 4 MULTIPLE INTEGRALS

CHAPTE 4 MULTIPLE INTEGAL The objects of this chpter re five-fold. They re: (1 Discuss when sclr-vlued functions f cn be integrted over closed rectngulr boxes in n ; simply put, f is integrble over iff

More information

### THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES

THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES 1. TRANSLATION Figure shows rigid body trnslting in three-dimensionl spce. Any two points in the body, such s A nd B, will move long prllel stright lines if

More information

### How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=

More information

### Review of Calculus, cont d

Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

More information

### Mathematics. Area under Curve.

Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding

More information

### Math 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008

Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 2-5pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of

More information

### ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

More information

### Summary: Method of Separation of Variables

Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section

More information

### Before we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!

Nme: Algebr II Honors Pre-Chpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble

More information

### Notes on length and conformal metrics

Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued

More information

### MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1

MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further

More information

### g i fφdx dx = x i i=1 is a Hilbert space. We shall, henceforth, abuse notation and write g i f(x) = f

1. Appliction of functionl nlysis to PEs 1.1. Introduction. In this section we give little introduction to prtil differentil equtions. In prticulr we consider the problem u(x) = f(x) x, u(x) = x (1) where

More information

### State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies

Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response

More information

### Chapter 14. Matrix Representations of Linear Transformations

Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn

More information

### p-adic Egyptian Fractions

p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction

More information

### 1 Probability Density Functions

Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our

More information

### P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)

1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this

More information

### PHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS

PHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS To strt on tensor clculus, we need to define differentition on mnifold.a good question to sk is if the prtil derivtive of tensor tensor on mnifold?

More information

### ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil

More information

### The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).

The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples

More information

### How can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?

Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those

More information

### 3.1 Review of Sine, Cosine and Tangent for Right Angles

Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles 125 3.1 Review of Sine, osine nd Tngent for Right ngles The word trigonometry is derived from the Greek words trigon,

More information

### 63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

More information

### 13.4 Work done by Constant Forces

13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push

More information

### Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as

Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph

More information

### Section 14.3 Arc Length and Curvature

Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in

More information

### Lecture 1. Functional series. Pointwise and uniform convergence.

1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is

More information

### UNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3

UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,

More information

### 3. Vectors. Home Page. Title Page. Page 2 of 37. Go Back. Full Screen. Close. Quit

Rutgers University Deprtment of Physics & Astronomy 01:750:271 Honors Physics I Lecture 3 Pge 1 of 37 3. Vectors Gols: To define vector components nd dd vectors. To introduce nd mnipulte unit vectors.

More information

### Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230

Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given

More information

### Lecture 3: Curves in Calculus. Table of contents

Mth 348 Fll 7 Lecture 3: Curves in Clculus Disclimer. As we hve textook, this lecture note is for guidnce nd supplement only. It should not e relied on when prepring for exms. In this lecture we set up

More information

### Advanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004

Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when

More information

### Chapter 9 Definite Integrals

Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

More information

### I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3

2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is

More information

### We partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.

Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn

More information

### Exponentials - Grade 10 [CAPS] *

OpenStx-CNX module: m859 Exponentils - Grde 0 [CAPS] * Free High School Science Texts Project Bsed on Exponentils by Rory Adms Free High School Science Texts Project Mrk Horner Hether Willims This work

More information

### THE KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART I MULTIPLE CHOICE NO CALCULATORS 90 MINUTES

THE 08 09 KENNESW STTE UNIVERSITY HIGH SHOOL MTHEMTIS OMPETITION PRT I MULTIPLE HOIE For ech of the following questions, crefully blcken the pproprite box on the nswer sheet with # pencil. o not fold,

More information

### Indefinite Integral. Chapter Integration - reverse of differentiation

Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the

More information

### Theoretical foundations of Gaussian quadrature

Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of

More information

### Taylor Polynomial Inequalities

Tylor Polynomil Inequlities Ben Glin September 17, 24 Abstrct There re instnces where we my wish to pproximte the vlue of complicted function round given point by constructing simpler function such s polynomil

More information

### SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics

SCHOOL OF ENGINEERING & BUIL ENVIRONMEN Mthemtics An Introduction to Mtrices Definition of Mtri Size of Mtri Rows nd Columns of Mtri Mtri Addition Sclr Multipliction of Mtri Mtri Multipliction 7 rnspose

More information

### 3. Vectors. Vectors: quantities which indicate both magnitude and direction. Examples: displacemement, velocity, acceleration

Rutgers University Deprtment of Physics & Astronomy 01:750:271 Honors Physics I Lecture 3 Pge 1 of 57 3. Vectors Vectors: quntities which indicte both mgnitude nd direction. Exmples: displcemement, velocity,

More information

### Loudoun Valley High School Calculus Summertime Fun Packet

Loudoun Vlley High School Clculus Summertime Fun Pcket We HIGHLY recommend tht you go through this pcket nd mke sure tht you know how to do everything in it. Prctice the problems tht you do NOT remember!

More information

### 12 TRANSFORMING BIVARIATE DENSITY FUNCTIONS

1 TRANSFORMING BIVARIATE DENSITY FUNCTIONS Hving seen how to trnsform the probbility density functions ssocited with single rndom vrible, the next logicl step is to see how to trnsform bivrite probbility

More information

### NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.

NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with

More information

### The Wave Equation I. MA 436 Kurt Bryan

1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string

More information

### Optimization Lecture 1 Review of Differential Calculus for Functions of Single Variable.

Optimiztion Lecture 1 Review of Differentil Clculus for Functions of Single Vrible http://users.encs.concordi.c/~luisrod, Jnury 14 Outline Optimiztion Problems Rel Numbers nd Rel Vectors Open, Closed nd

More information

### A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

More information

### Conservation Law. Chapter Goal. 5.2 Theory

Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very

More information

### Lecture 2e Orthogonal Complement (pages )

Lecture 2e Orthogonl Complement (pges -) We hve now seen tht n orthonorml sis is nice wy to descrie suspce, ut knowing tht we wnt n orthonorml sis doesn t mke one fll into our lp. In theory, the process

More information

### US01CMTH02 UNIT Curvature

Stu mteril of BSc(Semester - I) US1CMTH (Rdius of Curvture nd Rectifiction) Prepred by Nilesh Y Ptel Hed,Mthemtics Deprtment,VPnd RPTPScience College US1CMTH UNIT- 1 Curvture Let f : I R be sufficiently

More information

### Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus

Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl

More information

### Energy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon

Phys463.nb 49 7 Energy Bnds Ref: textbook, Chpter 7 Q: Why re there insultors nd conductors? Q: Wht will hppen when n electron moves in crystl? In the previous chpter, we discussed free electron gses,

More information

### Coordinate geometry and vectors

MST124 Essentil mthemtics 1 Unit 5 Coordinte geometry nd vectors Contents Contents Introduction 4 1 Distnce 5 1.1 The distnce etween two points in the plne 5 1.2 Midpoints nd perpendiculr isectors 7 2

More information

### THE NUMBER CONCEPT IN GREEK MATHEMATICS SPRING 2009

THE NUMBER CONCEPT IN GREEK MATHEMATICS SPRING 2009 0.1. VII, Definition 1. A unit is tht by virtue of which ech of the things tht exist is clled one. 0.2. VII, Definition 2. A number is multitude composed

More information