Conservative fields and potential functions. (Sect. 16.3) The line integral of a vector field along a curve.
|
|
- Helen Fletcher
- 6 years ago
- Views:
Transcription
1 onservative fields and potential functions. (Sect. 16.3) eview: Line integral of a vector field. onservative fields. The line integral of conservative fields. Finding the potential of a conservative field. omments on eact differential forms. The line integral of a vector field along a curve. Definition The line integral of a vector-valued function F : D n n, with n = 2, 3, along the curve associated with the function r : t, t 1 ] D 3 is given b F dr = F(t) r (t) dt t F r emark: An equivalent epression is: r (t) F dr = F(t) t r (t) r (t) dt, s1 F dr = ˆF û ds, s where û = r (t(s)) r (t(s)), and ˆF = F(t(s)).
2 Work done b a force on a particle. Definition In the case that the vector field F : D n n, with n = 2, 3, represents a force acting on a particle with position function r : t, t 1 ] D 3, then the line integral W = F dr, is called the work done b the force on the particle. F r A projectile of mass m moving on the surface of Earth. The movement takes place on a plane, and F =, mg. W in the first half of the trajector, and W on the second half. onservative fields and potential functions. (Sect. 16.3) eview: Line integral of a vector field. onservative fields. The line integral of conservative fields. Finding the potential of a conservative field. omments on eact differential forms.
3 onservative fields. Definition A vector field F : D n n, with n = 2, 3, is called conservative iff there eists a scalar function f : D n, called potential function, such that F = f. F r A projectile of mass m moving on the surface of Earth. The movement takes place on a plane, and F =, mg. F = f, with f = mg. onservative fields. 1 Show that the vector field F = ( , 2, 3 is )3/2 conservative and find the potential function. Solution: The field F = F 1, F 2, F 3 is conservative iff there eists a potential function f such that F = f, that is, F 1 = 1 f, F 2 = 2 f, F 3 = 3 f. Since i ( ( = 2 )3/2 i ) ] 1/2, i = 1, 2, 3, then we conclude that F = f, with f =
4 onservative fields and potential functions. (Sect. 16.3) eview: Line integral of a vector field. onservative fields. The line integral of conservative fields. Finding the potential of a conservative field. omments on eact differential forms. The line integral of conservative fields. Definition A set D n, with n = 2, 3, is called simpl connected iff ever two points in D can be connected b a smooth curve inside D and ever loop in D can be smoothl contracted to a point without leaving D. emark: A set is simpl connected iff it consists of one piece and it contains no holes. D Simpl connected Not simpl connected D
5 The line integral of conservative fields. Notation: If the path n, with n = 2, 3, has end points r, r 1, then denote the line integral of a field F along as follows r1 F dr = F dr. (This notation emphasizes the end points, not the path.) Theorem A smooth vector field F : D n n, with n = 2, 3, defined on a simpl connected domain D n is conservativewith F = f iff for ever two points r, r 1 D the line integral F dr is independent of the path joining r to r 1 and holds r1 F dr = f (r 1 ) f (r ). r emark: A field F is conservative iff F dr is path independent. r The line integral of conservative fields. Summar: F = f equivalent to Proof: Onl ( ). r1 r F dr = r1 r f dr = r1 r F dr = f (r 1 ) f (r ). where r(t ) = r and r(t 1 ) = r 1. Therefore, ( f ) r (t) dt, t r(t) r1 r F dr = t d ] f (r(t) dt = f (r(t1 )) f (r(t )). dt We conclude that r1 r F dr = f (r 1 ) f (r ). (The statement ( ) is more complicated to prove.)
6 The line integral of conservative fields. Evaluate I = (1,2,3) (,,) 2 d + 2 d + 2z dz. Solution: I is a line integral for a field in 3, since I = (1,2,3) (,,) 2, 2, 2z d, d, dz. Introduce F = 2, 2, 2z, r = (,, ) and r 1 = (1, 2, 3), then I = r1 r F dr. The field F is conservative, since F = f with potential f (,, z) = z 2. That is f (r) = r 2. Therefore, I = r1 r f dr = f (r 1 ) f (r ) = r 1 2 r 2 = ( ). We conclude that I = 14. The line integral of conservative fields. (Along a path.) Evaluate I = (1,2,3) (,,) 2 d + 2 d + 2z dz along a straight line. Solution: onsider the path given b r(t) = 1, 2, 3 t. Then r() =,,, and r(1) = 1, 2, 3. We now evaluate F = 2, 2, 2z along r(t), that is, F(t) = 2t, 4t, 6t. Therefore, I = t F(t) r (t) dt = 1 2t, 4t, 6t 1, 2, 3 dt I = 1 (2t + 8t + 18t) dt = 1 ( t 2 28t dt = ). We conclude that I = 14.
7 onservative fields and potential functions. (Sect. 16.3) eview: Line integral of a vector field. onservative fields. The line integral of conservative fields. Finding the potential of a conservative field. omments on eact differential forms. Finding the potential of a conservative field. Theorem (haracterization of potential fields) A smooth field F = F 1, F 2, F 3 on a simpl connected domain D 3 is a conservative field iff hold 2 F 3 = 3 F 2, 3 F 1 = 1 F 3, 1 F 2 = 2 F 1. Proof: Onl ( ). Since the vector field F is conservative, there eists a scalar field f such that F = f. Then the equations above are satisfied, since for i, j = 1, 2, 3 hold F i = i f i F j = i j f = j i f = j F i. (The statement ( ) is more complicated to prove.)
8 Finding the potential of a conservative field. Show that the field F = 2, ( 2 z 2 ), 2z is conservative. Solution: We need to show that the equations in the Theorem above hold, that is 2 F 3 = 3 F 2, 3 F 1 = 1 F 3, 1 F 2 = 2 F 1. with 1 =, 2 =, and 3 = z. This is the case, since 1 F 2 = 2, 2 F 1 = 2, 2 F 3 = 2z, 3 F 2 = 2z, 3 F 1 =, 1 F 3 =. Finding the potential of a conservative field. Find the potential function of the conservative field F = 2, ( 2 z 2 ), 2z. Solution: We know there eists a scalar function f solution of F = f f = 2, f = 2 z 2, z f = 2z. f = 2 d + g(, z) f = 2 + g(, z). f = 2 + g(, z) = 2 z 2 g(, z) = z 2. g(, z) = z 2 d +h(z) = z 2 +h(z) f = 2 z 2 +h(z). z f = 2z + z h(z) = 2z z h(z) = f = ( 2 z 2 ) +c.
9 onservative fields and potential functions. (Sect. 16.3) eview: Line integral of a vector field. onservative fields. The line integral of conservative fields. Finding the potential of a conservative field. omments on eact differential forms. omments on eact differential forms. Notation: We call a differential form to the integrand in a line integral for a smooth field F, that is, F dr = F, F, F z d, d, dz = F d + F d + F z dz. emark: A differential form is a quantit that can be integrated along a path. Definition A differential form F dr = F d + F d + F z dz is called eact iff there eists a scalar function f such that emarks: F d + F d + F z dz = f d + f d + z f dz. A differential form F dr is eact iff F = f. An eact differential form is nothing else than another name for a conservative field.
10 omments on eact differential forms. Show that the differential form given below is eact, where F dr = 2 d + ( 2 z 2 ) d 2z dz. Solution: We need to do the same calculation we did above: Writing F dr = F 1 d 1 + F 2 d 2 + F 3 d 3, show that 2 F 3 = 3 F 2, 3 F 1 = 1 F 3, 1 F 2 = 2 F 1. with 1 =, 2 =, and 3 = z. We showed that this is the case, since 1 F 2 = 2, 2 F 1 = 2, 2 F 3 = 2z, 3 F 2 = 2z, 3 F 1 =, 1 F 3 =. So, there eists f such that F dr = f dr. Green s Theorem on a plane. (Sect. 16.4) eview: Line integrals and flu integrals. Green s Theorem on a plane. irculation-tangential form. Flu-normal form. Tangential and normal forms equivalence.
11 eview: The line integral of a vector field along a curve. Definition The line integral of a vector-valued function F : D n n, with n = 2, 3, along the curve r : t, t 1 ] D 3, with arc length function s, is given b s1 F u ds = F(t) r (t) dt, s t where u = r r, and s = s(t ), s 1 = s(t 1 ). z u { z = } emark: Since F = F, F and r(t) = (t), (t), in components, = t t F(t) r (t) dt F (t) (t) + F (t) (t) ] dt. eview: The line integral of a vector field along a curve. Evaluate the line integral of F =, along the loop r(t) = cos(t), sin(t) for t, 2π]. Solution: Evaluate F along the curve: F(t) = sin(t), cos(t). Now compute the derivative vector r (t) = sin(t), cos(t). Then evaluate the line integral in components, F u ds = F (t) (t) + F (t) (t) ] dt, F u ds = F u ds = 2π 2π t ( sin(t))( sin(t)) + cos(t) cos(t) ] dt, sin 2 (t) + cos 2 (t) ] dt F u ds = 2π.
12 eview: The flu across a plane loop. Definition The flu of a vector field F : {z = } 3 {z = } 3 along a closed plane loop r : t, t 1 ] {z = } 3 is given b F = F n ds, where n is the unit outer normal vector to the curve inside the plane {z = }. z n { z = } emark: Since F = F, F,, r(t) = (t), (t),, ds = r (t) dt, and n = 1 r (t), (t),, in components, F n ds = F (t) (t) F (t) (t) ] dt. t eview: The flu across a plane loop. Evaluate the flu of F =,, along the loop r(t) = cos(t), sin(t), for t, 2π]. Solution: Evaluate F along the curve: F(t) = sin(t), cos(t),. Now compute the derivative vector r (t) = sin(t), cos(t),. Now compute the normal vector n(t) = (t), (t),, that is, n(t) = cos(t), sin(t),. Evaluate the flu integral in components, F n ds = F n ds = 2π F u ds = t F (t) (t) F (t) (t) ] dt, sin(t) cos(t) cos(t)( sin(t)) ] dt, 2π dt F u ds =.
13 Green s Theorem on a plane. (Sect. 16.4) eview: Line integrals and flu integrals. Green s Theorem on a plane. irculation-tangential form. Flu-normal form. Tangential and normal forms equivalence. Green s Theorem on a plane. Theorem (irculation-tangential form) The counterclockwise line integral F u ds of the field F = F, F along a loop enclosing a region 2 and given b the function r(t) = (t), (t) for t t, t 1 ] and with unit tangent vector u, satisfies that F (t) (t) + F (t) (t) ] dt = F F d d. t z u { z = } Equivalentl, F u ds = ( F F ) d d.
14 Green s Theorem on a plane. Verif Green s Theorem tangential form for the field F =, and the loop r(t) = cos(t), sin(t) for t, 2π]. Solution: ecall: We found that F u ds = 2π. Now we compute the double integral I = F F d d and we verif that we get the same result, 2π. I = 1 ( 1) ] d d = 2 2π 1 d d = 2 r dr dθ ( r 2 I = 2(2π) 1 ) I = 2π. 2 We verified that F u ds = F F d d = 2π. Green s Theorem on a plane. (Sect. 16.4) eview: Line integrals and flu integrals. Green s Theorem on a plane. irculation-tangential form. Flu-normal form. Tangential and normal forms equivalence.
15 Green s Theorem on a plane. Theorem (Flu-normal form) The counterclockwise flu integral F n ds of the field F = F, F along a loop enclosing a region 2 and given b the function r(t) = (t), (t) for t t, t 1 ] and with unit normal vector n, satisfies that F (t) (t) F (t) (t) ] dt = F + F d d. t z n { z = } Equivalentl, F n ds = ( F + F ) d d. Green s Theorem on a plane. Verif Green s Theorem normal form for the field F =, and the loop r(t) = cos(t), sin(t) for t, 2π]. Solution: ecall: We found that F n ds =. Now we compute the double integral I = F + F d d and we verif that we get the same result,. I = ( ) + () ] d d = We verified that d d =. F n ds = F + F d d =.
16 Green s Theorem on a plane. Verif Green s Theorem normal form for the field F = 2, 3 and the loop r(t) = a cos(t), a sin(t) for t, 2π], a >. Solution: We start with the line integral F n ds = F (t) (t) F (t) (t) ] dt. t It is simple to see that F(t) = 2a cos(t), 3a sin(t), and also that r (t) = a sin(t), a cos(t). 2π Therefore, F n ds = 2a 2 cos 2 (t) 3a 2 sin 2 (t) ] dt, 2π F n ds = 2a cos(2t) 3a 2 1 ] 1 cos(2t) dt π Since cos(2t) dt =, we conclude F n ds = πa 2. Green s Theorem on a plane. Verif Green s Theorem normal form for the field F = 2, 3 and the loop r(t) = a cos(t), a sin(t) for t, 2π], a >. Solution: ecall: F n ds = πa 2. Now we compute the double integral I = F + F d d. I = I = Hence, (2) + ( 3) ] d d = 2π d d = a (2 3) d d. ( r 2 r dr dθ = 2π a ) = πa 2. 2 F n ds = F + F d d = πa 2.
17 Green s Theorem on a plane. (Sect. 16.4) eview: Line integrals and flu integrals. Green s Theorem on a plane. irculation-tangential form. Flu-normal form. Tangential and normal forms equivalence. Tangential and normal forms equivalence. Lemma The Green Theorem in tangential form is equivalent to the Green Theorem in normal form. Proof: Green s Theorem in tangential form for F = F, F sas F (t) (t) + F (t) (t) ] dt = F F d d. t Appl this Theorem for ˆF = F, F, that is, ˆF = F and ˆF = F. We obtain F (t) (t) + F (t) (t) ] ( dt = F ( F ) ) d d, t so, F (t) (t) F (t) (t) ] dt = F + F d d, t which is Green s Theorem in normal form. The converse implication is proved in the same wa.
18 Using Green s Theorem Use Green s Theorem to find the counterclockwise circulation of the field F = ( 2 2 ), ( ) along the curve that is the triangle bounded b =, = 3 and =. Solution: ecall: F dr = F F d d. F dr = F dr = 3 F dr = (2 2) d d = ( ) ( 2 )] d = 2 d = (2 2) d d, 3 ( 2 2 2) d, F dr = 9. Green s Theorem on a plane. (Sect. 16.4) eview of Green s Theorem on a plane. Sketch of the proof of Green s Theorem. Divergence and curl of a function on a plane. Area computed with a line integral.
19 eview: Green s Theorem on a plane. Theorem Given a field F = F, F and a loop enclosing a region 2 described b the function r(t) = (t), (t) for t t, t 1 ], with unit tangent vector u and eterior normal vector n, then holds: The counterclockwise line integral F u ds satisfies: F (t) (t) + F (t) (t) ] dt = t The counterclockwise line integral F (t) (t) F (t) (t) ] dt = t ( F F ) d d. F n ds satisfies: ( F + F ) d d. eview: Green s Theorem on a plane. z z u n { z = } { z = } irculation-tangential form: F u ds = F F d d. Flu-normal form: F n ds = F + F d d. Lemma The Green Theorem in tangential form is equivalent to the Green Theorem in normal form.
20 Green s Theorem on a plane. (Sect. 16.4) eview of Green s Theorem on a plane. Sketch of the proof of Green s Theorem. Divergence and curl of a function on a plane. Area computed with a line integral. Sketch of the proof of Green s Theorem. We want to prove that for ever differentiable vector field F = F, F the Green Theorem in tangential form holds, F (t) (t) + F (t) (t) ] dt = F F d d. We onl consider a simple domain like the one in the pictures. = g () 1 1 = h () = g () 1 Using the picture on the left we show that F (t) (t) dt = F d d; and using the picture on the right we show that F (t) (t) dt = F d d. = h () 1
21 Sketch of the proof of Green s Theorem. = g () 1 = g () 1 Show that for F (t) = F ((t), (t)) holds F (t) (t) dt = F d d; The path can be described b the curves r and r 1 given b Therefore, r (t) = t, g (t), t, 1 ] r 1 (t) = ( 1 + t), g 1 ( 1 + t) t, 1 ]. r (t) = 1, g (t), t, 1 ] r 1(t) = 1, g 1( 1 + t) t, 1 ]. ecall: F (t) = F (t, g (t)) on r, and F (t) = F (( 1 + t), g 1 ( 1 + t)) on r 1. Sketch of the proof of Green s Theorem. 1 F (t) (t) dt = 1 F (t, g (t)) dt F (( 1 + t), g 1 ( 1 + t)) dt Substitution in the second term: τ = 1 + t, so dτ = dt. 1 1 F (( 1 + t), g 1 ( 1 + t)) dt = F (τ, g 1 (τ)) ( dτ) = Therefore, F (t) (t) dt = We obtain: F (t) (t) dt = F (τ, g 1 (τ)) dτ. F (t, g (t)) F (t, g 1 (t)) ] dt. g1 (t) g (t) F (t, ) ] d dt.
22 Sketch of the proof of Green s Theorem. ecall: F (t) (t) dt = 1 g1 (t) g (t) F (t, ) ] d dt. This result is precisel what we wanted to prove: F (t) (t) dt = F d d. We just mention that the result F (t) (t) dt = F d d. 1 = h () is proven in a similar wa using the parametrization of the given in the picture. = h () 1 Green s Theorem on a plane. (Sect. 16.4) eview of Green s Theorem on a plane. Sketch of the proof of Green s Theorem. Divergence and curl of a function on a plane. Area computed with a line integral.
23 Divergence and curl of a function on a plane. Definition The curl of a vector field F = F, F in 2 is the scalar ( curl F ) z = F F. The divergence of a vector field F = F, F in 2 is the scalar div F = F + F. emark: Both forms of Green s Theorem can be written as: F u ds = curl F d d. F n ds = z div F d d. Divergence and curl of a function on a plane. emark: What tpe of information about F is given in ( curl F ) z? : Suppose F is the velocit field of a viscous fluid and F =, ( curl F ) z = F F = 2. If we place a small ball at (, ), the ball will spin around the z-ais with speed proportional to ( curl F ) z. If we place a small ball at everwhere in the plane, the ball will spin around the z-ais with speed proportional to ( curl F ) z. emark: The curl of a field measures its rotation.
24 Divergence and curl of a function on a plane. emark: What tpe of information about F is given in div F? : Suppose F is the velocit field of a gas and F =, div F = F + F = 2. The field F represents the gas as is heated with a heat source at (, ). The heated gas epands in all directions, radiall out form (, ). The div F measures that epansion. emark: The divergence of a field measures its epansion. emarks: Notice that for F =, we have ( curl F ) z =. Notice that for F =, we have div F =. Green s Theorem on a plane. (Sect. 16.4) eview of Green s Theorem on a plane. Sketch of the proof of Green s Theorem. Divergence and curl of a function on a plane. Area computed with a line integral.
25 Area computed with a line integral. emark: An of the two versions of Green s Theorem can be used to compute areas using a line integral. For eample: ( F + F d d = F d F d) If F is such that the left-hand side above has integrand 1, then that integral is the area A() of the region. Indeed: F =, d d = A() = d. F =, d d = A() = d. F = 1 2, d d = A() = 1 2 ( d d ). Area computed with a line integral. Use Green s Theorem to find the area of the region enclosed b the ellipse r(t) = a cos(t), b sin(t), with t, 2π] and a, b positive. Solution: We use: A() = d. We need to compute r (t) = a sin(t), b cos(t). Then, A() = 2π (t) (t) dt = 2π a cos(t) b cos(t) dt. Since A() = ab 2π 2π cos 2 (t) dt = ab 2π cos(2t) dt =, we obtain A() = ab 2 A() = πab. 1 ] 1 + cos(2t) dt. 2 2π, that is,
Green s Theorem Jeremy Orloff
Green s Theorem Jerem Orloff Line integrals and Green s theorem. Vector Fields Vector notation. In 8.4 we will mostl use the notation (v) = (a, b) for vectors. The other common notation (v) = ai + bj runs
More informationDirectional derivatives and gradient vectors (Sect. 14.5) Directional derivative of functions of two variables.
Directional derivatives and gradient vectors (Sect. 14.5) Directional derivative of functions of two variables. Partial derivatives and directional derivatives. Directional derivative of functions of three
More informationis the two-dimensional curl of the vector field F = P, Q. Suppose D is described by a x b and f(x) y g(x) (aka a type I region).
Math 55 - Vector alculus II Notes 4.4 Green s Theorem We begin with Green s Theorem: Let be a positivel oriented (parameterized counterclockwise) piecewise smooth closed simple curve in R and be the region
More informationSections minutes. 5 to 10 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed.
MTH 34 Review for Exam 4 ections 16.1-16.8. 5 minutes. 5 to 1 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed. Review for Exam 4 (16.1) Line
More informationES.182A Topic 41 Notes Jeremy Orloff. 41 Extensions and applications of Green s theorem
ES.182A Topic 41 Notes Jerem Orloff 41 Etensions and applications of Green s theorem 41.1 eview of Green s theorem: Tangential (work) form: F T ds = curlf d d M d + N d = N M d d. Normal (flu) form: F
More information(0,2) L 1 L 2 R (-1,0) (2,0) MA4006: Exercise Sheet 3: Solutions. 1. Evaluate the integral R
MA6: Eercise Sheet 3: Solutions 1. Evaluate the integral d d over the triangle with vertices ( 1, ), (, 2) and (2, ). Solution. See Figure 1. Let be the inner variable and the outer variable. we need the
More informationIntegrals along a curve in space. (Sect. 16.1)
Integrals along a curve in space. (Sect. 6.) Line integrals in space. The addition of line integrals. ass and center of mass of wires. Line integrals in space Definition The line integral of a function
More informationMATH Line integrals III Fall The fundamental theorem of line integrals. In general C
MATH 255 Line integrals III Fall 216 In general 1. The fundamental theorem of line integrals v T ds depends on the curve between the starting point and the ending point. onsider two was to get from (1,
More informationVector-Valued Functions
Vector-Valued Functions 1 Parametric curves 8 ' 1 6 1 4 8 1 6 4 1 ' 4 6 8 Figure 1: Which curve is a graph of a function? 1 4 6 8 1 8 1 6 4 1 ' 4 6 8 Figure : A graph of a function: = f() 8 ' 1 6 4 1 1
More informationOne side of each sheet is blank and may be used as scratch paper.
Math 244 Spring 2017 (Practice) Final 5/11/2017 Time Limit: 2 hours Name: No calculators or notes are allowed. One side of each sheet is blank and may be used as scratch paper. heck your answers whenever
More information49. Green s Theorem. The following table will help you plan your calculation accordingly. C is a simple closed loop 0 Use Green s Theorem
49. Green s Theorem Let F(x, y) = M(x, y), N(x, y) be a vector field in, and suppose is a path that starts and ends at the same point such that it does not cross itself. Such a path is called a simple
More informationLine Integrals and Green s Theorem Jeremy Orloff
Line Integrals and Green s Theorem Jerem Orloff Vector Fields (or vector valued functions) Vector notation. In 8.4 we will mostl use the notation (v) = (a, b) for vectors. The other common notation (v)
More informationMath 107. Rumbos Fall Solutions to Review Problems for Exam 3
Math 17. umbos Fall 29 1 Solutions to eview Problems for Eam 3 1. Consider a wheel of radius a which is rolling on the ais in the plane. Suppose that the center of the wheel moves in the positive direction
More informationVector Calculus. Vector Fields. Reading Trim Vector Fields. Assignment web page assignment #9. Chapter 14 will examine a vector field.
Vector alculus Vector Fields Reading Trim 14.1 Vector Fields Assignment web page assignment #9 hapter 14 will eamine a vector field. For eample, if we eamine the temperature conditions in a room, for ever
More information18.02 Review Jeremy Orloff. 1 Review of multivariable calculus (18.02) constructs
18.02 eview Jerem Orloff 1 eview of multivariable calculus (18.02) constructs 1.1 Introduction These notes are a terse summar of what we ll need from multivariable calculus. If, after reading these, some
More informationMATH 223 FINAL EXAM STUDY GUIDE ( )
MATH 3 FINAL EXAM STUDY GUIDE (017-018) The following questions can be used as a review for Math 3 These questions are not actual samples of questions that will appear on the final eam, but the will provide
More informationMath 212-Lecture 20. P dx + Qdy = (Q x P y )da. C
15. Green s theorem Math 212-Lecture 2 A simple closed curve in plane is one curve, r(t) : t [a, b] such that r(a) = r(b), and there are no other intersections. The positive orientation is counterclockwise.
More informationPractice Problems for Exam 3 (Solutions) 1. Let F(x, y) = xyi+(y 3x)j, and let C be the curve r(t) = ti+(3t t 2 )j for 0 t 2. Compute F dr.
1. Let F(x, y) xyi+(y 3x)j, and let be the curve r(t) ti+(3t t 2 )j for t 2. ompute F dr. Solution. F dr b a 2 2 F(r(t)) r (t) dt t(3t t 2 ), 3t t 2 3t 1, 3 2t dt t 3 dt 1 2 4 t4 4. 2. Evaluate the line
More informationIntroduction to Differential Equations
Math0 Lecture # Introduction to Differential Equations Basic definitions Definition : (What is a DE?) A differential equation (DE) is an equation that involves some of the derivatives (or differentials)
More informationContents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9
MATH 32B-2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B-2 (8W)
More informationScalar functions of several variables (Sect. 14.1)
Scalar functions of several variables (Sect. 14.1) Functions of several variables. On open, closed sets. Functions of two variables: Graph of the function. Level curves, contour curves. Functions of three
More informationMath Review for Exam 3
1. ompute oln: (8x + 36xy)ds = Math 235 - Review for Exam 3 (8x + 36xy)ds, where c(t) = (t, t 2, t 3 ) on the interval t 1. 1 (8t + 36t 3 ) 1 + 4t 2 + 9t 4 dt = 2 3 (1 + 4t2 + 9t 4 ) 3 2 1 = 2 3 ((14)
More informationMTH234 Chapter 16 - Vector Calculus Michigan State University
MTH234 hapter 6 - Vector alculus Michigan State Universit 4 Green s Theorem Green s Theorem gives a relationship between double integrals and line integrals around simple closed curves. (Start and end
More informationDirectional derivatives and gradient vectors (Sect. 14.5). Directional derivative of functions of two variables.
Directional derivatives and gradient vectors (Sect. 14.5). Directional derivative of functions of two variables. Partial derivatives and directional derivatives. Directional derivative of functions of
More informationES.182A Topic 36 Notes Jeremy Orloff
ES.82A Topic 36 Notes Jerem Orloff 36 Vector fields and line integrals in the plane 36. Vector analsis We now will begin our stud of the part of 8.2 called vector analsis. This is the stud of vector fields
More informationENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 3 Solutions [Multiple Integration; Lines of Force]
ENGI 44 Advanced Calculus for Engineering Facult of Engineering and Applied Science Problem Set Solutions [Multiple Integration; Lines of Force]. Evaluate D da over the triangular region D that is bounded
More informationMATH 2400 Final Exam Review Solutions
MATH Final Eam eview olutions. Find an equation for the collection of points that are equidistant to A, 5, ) and B6,, ). AP BP + ) + y 5) + z ) 6) y ) + z + ) + + + y y + 5 + z 6z + 9 + 6 + y y + + z +
More informationMcGill University April 16, Advanced Calculus for Engineers
McGill University April 16, 2014 Faculty of cience Final examination Advanced Calculus for Engineers Math 264 April 16, 2014 Time: 6PM-9PM Examiner: Prof. R. Choksi Associate Examiner: Prof. A. Hundemer
More informationPhysics 207, Lecture 4, Sept. 15
Phsics 07, Lecture 4, Sept. 15 Goals for hapts.. 3 & 4 Perform vector algebra (addition & subtraction) graphicall or b, & z components Interconvert between artesian and Polar coordinates Distinguish position-time
More information(b) Find the range of h(x, y) (5) Use the definition of continuity to explain whether or not the function f(x, y) is continuous at (0, 0)
eview Exam Math 43 Name Id ead each question carefully. Avoid simple mistakes. Put a box around the final answer to a question (use the back of the page if necessary). For full credit you must show your
More informationC 3 C 4. R k C 1. (x,y)
16.4 1 16.4 Green s Theorem irculation Density (x,y + y) 3 (x+ x,y + y) 4 k 2 (x,y) 1 (x+ x,y) Suppose that F(x,y) M(x,y)i+N(x,y)j is the velocity field of a fluid flow in the plane and that the first
More informationD = 2(2) 3 2 = 4 9 = 5 < 0
1. (7 points) Let f(, ) = +3 + +. Find and classif each critical point of f as a local minimum, a local maimum, or a saddle point. Solution: f = + 3 f = 3 + + 1 f = f = 3 f = Both f = and f = onl at (
More informationMath 11 Fall 2007 Practice Problem Solutions
Math 11 Fall 27 Practice Problem olutions Here are some problems on the material we covered since the second midterm. This collection of problems is not intended to mimic the final in length, content,
More informationLine Integrals Independence of Path -(9.9) f x
. Differentials and Eact Differentials: The differentials of f, and f,,z are: Line Integrals Independence of Path -(9.9) d d, and d d dz. A differential epression P,,zd Q,,zd R,,zdz is said to be an eact
More informationFaculty of Engineering, Mathematics and Science School of Mathematics
Faculty of Engineering, Mathematics and Science School of Mathematics SF Engineers SF MSISS SF MEMS MAE: Engineering Mathematics IV Trinity Term 18 May,??? Sports Centre??? 9.3 11.3??? Prof. Sergey Frolov
More informationSOME PROBLEMS YOU SHOULD BE ABLE TO DO
OME PROBLEM YOU HOULD BE ABLE TO DO I ve attempted to make a list of the main calculations you should be ready for on the exam, and included a handful of the more important formulas. There are no examples
More informationVector Functions & Space Curves MATH 2110Q
Vector Functions & Space Curves Vector Functions & Space Curves Vector Functions Definition A vector function or vector-valued function is a function that takes real numbers as inputs and gives vectors
More informationAPPM 1360 Final Exam Spring 2016
APPM 36 Final Eam Spring 6. 8 points) State whether each of the following quantities converge or diverge. Eplain your reasoning. a) The sequence a, a, a 3,... where a n ln8n) lnn + ) n!) b) ln d c) arctan
More informationExtra Problems Chapter 7
MA11: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 1 Etra Problems hapter 7 1. onsider the vector field F = i+z j +z 3 k. a) ompute div F. b) ompute curl F. Solution a) div F = +z +3z b) curl F = i
More informationExtra Problems Chapter 7
MA11: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 1 Etra Problems hapter 7 1. onsider the vector field F = i+z j +z 3 k. a) ompute div F. b) ompute curl F. Solution a) div F = +z +3z b) curl F = i
More informationCartesian coordinates in space (Sect. 12.1).
Cartesian coordinates in space (Sect..). Overview of Multivariable Calculus. Cartesian coordinates in space. Right-handed, left-handed Cartesian coordinates. Distance formula between two points in space.
More informationFunctions of Several Variables
Chapter 1 Functions of Several Variables 1.1 Introduction A real valued function of n variables is a function f : R, where the domain is a subset of R n. So: for each ( 1,,..., n ) in, the value of f is
More information7a3 2. (c) πa 3 (d) πa 3 (e) πa3
1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin
More informationDivergence Theorem December 2013
Divergence Theorem 17.3 11 December 2013 Fundamental Theorem, Four Ways. b F (x) dx = F (b) F (a) a [a, b] F (x) on boundary of If C path from P to Q, ( φ) ds = φ(q) φ(p) C φ on boundary of C Green s Theorem:
More informationArc-length of a curve on the plane (Sect. 11.2) Review: Parametric curves on the plane
Arc-length of a curve on the plane (Sect. 11.2) Review: Parametric curves on the plane. The slope of tangent lines to curves. The arc-length of a curve. The arc-length function and differential. Review:
More informationHOMEWORK 2 SOLUTIONS
HOMEWORK SOLUTIONS MA11: ADVANCED CALCULUS, HILARY 17 (1) Find parametric equations for the tangent line of the graph of r(t) = (t, t + 1, /t) when t = 1. Solution: A point on this line is r(1) = (1,,
More informationDepartment of Mathematical and Statistical Sciences University of Alberta
MATH 4 (R) Winter 8 Intermediate Calculus I Solutions to Problem Set #5 Completion Date: Frida Februar 5, 8 Department of Mathematical and Statistical Sciences Universit of Alberta Question. [Sec.., #
More informationDivergence Theorem Fundamental Theorem, Four Ways. 3D Fundamental Theorem. Divergence Theorem
Divergence Theorem 17.3 11 December 213 Fundamental Theorem, Four Ways. b F (x) dx = F (b) F (a) a [a, b] F (x) on boundary of If C path from P to Q, ( φ) ds = φ(q) φ(p) C φ on boundary of C Green s Theorem:
More informationMath 11 Fall 2016 Final Practice Problem Solutions
Math 11 Fall 216 Final Practice Problem olutions Here are some problems on the material we covered since the second midterm. This collection of problems is not intended to mimic the final in length, content,
More informationMath 234 Exam 3 Review Sheet
Math 234 Exam 3 Review Sheet Jim Brunner LIST OF TOPIS TO KNOW Vector Fields lairaut s Theorem & onservative Vector Fields url Divergence Area & Volume Integrals Using oordinate Transforms hanging the
More informationMath 21a: Multivariable calculus. List of Worksheets. Harvard University, Spring 2009
Math 2a: Multivariable calculus Harvard Universit, Spring 2009 List of Worksheets Vectors and the Dot Product Cross Product and Triple Product Lines and Planes Functions and Graphs Quadric Surfaces Vector-Valued
More informationMAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative.
MAC2313 Final A (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. ii. The vector field F = 5(x 2 + y 2 ) 3/2 x, y is radial. iii. All constant
More information13.1. For further details concerning the physics involved and animations of the trajectories of the particles, see the following websites:
8 CHAPTER VECTOR FUNCTIONS N Some computer algebra sstems provide us with a clearer picture of a space curve b enclosing it in a tube. Such a plot enables us to see whether one part of a curve passes in
More informationDIFFERENTIATION. 3.1 Approximate Value and Error (page 151)
CHAPTER APPLICATIONS OF DIFFERENTIATION.1 Approimate Value and Error (page 151) f '( lim 0 f ( f ( f ( f ( f '( or f ( f ( f '( f ( f ( f '( (.) f ( f '( (.) where f ( f ( f ( Eample.1 (page 15): Find
More informationFinal Review Worksheet
Score: Name: Final Review Worksheet Math 2110Q Fall 2014 Professor Hohn Answers (in no particular order): f(x, y) = e y + xe xy + C; 2; 3; e y cos z, e z cos x, e x cos y, e x sin y e y sin z e z sin x;
More informationMultivariable Calculus Lecture #13 Notes. in each piece. Then the mass mk. 0 σ = σ = σ
Multivariable Calculus Lecture #1 Notes In this lecture, we ll loo at parameterization of surfaces in R and integration on a parameterized surface Applications will include surface area, mass of a surface
More informationGreen s, Divergence, Stokes: Statements and First Applications
Math 425 Notes 12: Green s, Divergence, tokes: tatements and First Applications The Theorems Theorem 1 (Divergence (planar version)). Let F be a vector field in the plane. Let be a nice region of the plane
More informationCURRENT MATERIAL: Vector Calculus.
Math 275, section 002 (Ultman) Spring 2012 FINAL EXAM REVIEW The final exam will be held on Wednesday 9 May from 8:00 10:00am in our regular classroom. You will be allowed both sides of two 8.5 11 sheets
More informationFinal exam (practice 1) UCLA: Math 32B, Spring 2018
Instructor: Noah White Date: Final exam (practice 1) UCLA: Math 32B, Spring 218 This exam has 7 questions, for a total of 8 points. Please print your working and answers neatly. Write your solutions in
More informationVector Calculus. Chapter Vector Fields A vector field describes the motion of the air as a vector at each point in the room
hapter 5 Vector alculus 5. Vector Fields 5.. A vector field describes the motion of the air as a vector at each point in the room. 5.. y 4 4 4 4 5..3 At selected points a, b, plot the vector fa, b,ga,
More informationMATH 332: Vector Analysis Summer 2005 Homework
MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,
More information(a) We split the square up into four pieces, parametrizing and integrating one a time. Right side: C 1 is parametrized by r 1 (t) = (1, t), 0 t 1.
Thursda, November 5 Green s Theorem Green s Theorem is a 2-dimensional version of the Fundamental Theorem of alculus: it relates the (integral of) a vector field F on the boundar of a region to the integral
More informationLecture 04. Curl and Divergence
Lecture 04 Curl and Divergence UCF Curl of Vector Field (1) F c d l F C Curl (or rotor) of a vector field a n curlf F d l lim c s s 0 F s a n C a n : normal direction of s follow right-hand rule UCF Curl
More informationYing-Ying Tran 2016 May 10 Review
MATH 920 Final review Ying-Ying Tran 206 Ma 0 Review hapter 3: Vector geometr vectors dot products cross products planes quadratic surfaces clindrical and spherical coordinates hapter 4: alculus of vector-valued
More informationAPJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE EXAMINATION, FEBRUARY 2017 MA101: CALCULUS PART A
A B1A003 Pages:3 (016 ADMISSIONS) Reg. No:... Name:... APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE EXAMINATION, FEBRUARY 017 MA101: CALCULUS Ma. Marks: 100 Duration: 3 Hours PART
More informationArc Length and Curvature
Arc Length and Curvature. Last time, we saw that r(t) = cos t, sin t, t parameteried the pictured curve. (a) Find the arc length of the curve between (, 0, 0) and (, 0, π). (b) Find the unit tangent vector
More informationSome Important Concepts and Theorems of Vector Calculus
ome Important oncepts and Theorems of Vector alculus 1. The Directional Derivative. If f(x) is a scalar-valued function of, say x, y, and z, andu is a direction (i.e. a unit vector), then the Directional
More informationVector Calculus, Maths II
Section A Vector Calculus, Maths II REVISION (VECTORS) 1. Position vector of a point P(x, y, z) is given as + y and its magnitude by 2. The scalar components of a vector are its direction ratios, and represent
More informationExercises of Mathematical analysis II
Eercises of Mathematical analysis II In eercises. - 8. represent the domain of the function by the inequalities and make a sketch showing the domain in y-plane.. z = y.. z = arcsin y + + ln y. 3. z = sin
More informationMajor Ideas in Calc 3 / Exam Review Topics
Major Ideas in Calc 3 / Exam Review Topics Here are some highlights of the things you should know to succeed in this class. I can not guarantee that this list is exhaustive!!!! Please be sure you are able
More informationMATHEMATICS 317 December 2010 Final Exam Solutions
MATHEMATI 317 December 1 Final Eam olutions 1. Let r(t) = ( 3 cos t, 3 sin t, 4t ) be the position vector of a particle as a function of time t. (a) Find the velocity of the particle as a function of time
More informationSummary of various integrals
ummary of various integrals Here s an arbitrary compilation of information about integrals Moisés made on a cold ecember night. 1 General things o not mix scalars and vectors! In particular ome integrals
More informationCalculus III. Math 233 Spring Final exam May 3rd. Suggested solutions
alculus III Math 33 pring 7 Final exam May 3rd. uggested solutions This exam contains twenty problems numbered 1 through. All problems are multiple choice problems, and each counts 5% of your total score.
More information3 Applications of Derivatives Instantaneous Rates of Change Optimization Related Rates... 13
Contents Limits Derivatives 3. Difference Quotients......................................... 3. Average Rate of Change...................................... 4.3 Derivative Rules...........................................
More informationMath 240 Calculus III
Line Math 114 Calculus III Summer 2013, Session II Monday, July 1, 2013 Agenda Line 1.,, and 2. Line vector Line Definition Let f : X R 3 R be a differentiable scalar function on a region of 3-dimensional
More informationENGI 4430 Line Integrals; Green s Theorem Page 8.01
ENGI 4430 Line Integrals; Green s Theorem Page 8.01 8. Line Integrals Two applications of line integrals are treated here: the evaluation of work done on a particle as it travels along a curve in the presence
More informationIntegration in the Complex Plane (Zill & Wright Chapter 18)
Integration in the omplex Plane Zill & Wright hapter 18) 116-4-: omplex Variables Fall 11 ontents 1 ontour Integrals 1.1 Definition and Properties............................. 1. Evaluation.....................................
More information4.3 Mean-Value Theorem and Monotonicity
.3 Mean-Value Theorem and Monotonicit 1. Mean Value Theorem Theorem: Suppose that f is continuous on the interval a, b and differentiable on the interval a, b. Then there eists a number c in a, b such
More informationCHAPTER 11 Vector-Valued Functions
CHAPTER Vector-Valued Functions Section. Vector-Valued Functions...................... 9 Section. Differentiation and Integration of Vector-Valued Functions.... Section. Velocit and Acceleration.....................
More informationy=1/4 x x=4y y=x 3 x=y 1/3 Example: 3.1 (1/2, 1/8) (1/2, 1/8) Find the area in the positive quadrant bounded by y = 1 x and y = x3
Eample: 3.1 Find the area in the positive quadrant bounded b 1 and 3 4 First find the points of intersection of the two curves: clearl the curves intersect at (, ) and at 1 4 3 1, 1 8 Select a strip at
More informationMATHEMATICS 317 April 2017 Final Exam Solutions
MATHEMATI 7 April 7 Final Eam olutions. Let r be the vector field r = îı + ĵj + z ˆk and let r be the function r = r. Let a be the constant vector a = a îı + a ĵj + a ˆk. ompute and simplif the following
More informationDr. F. Wilhelm, DVC, Work, path-integrals, vector operators. C:\physics\130 lecture-giancoli\ch 07&8a work pathint vect op.
C:\phsics\30 lecture-giancoli\ch 07&8a work pathint vect op.doc 4/3/0 of 4. CONCEPT OF WORK W:.... CONCEPT OF POTENTIL ENERGY U:... 3. Force of a Spring... 3 4. Force with several variables.... 4 5. Eamples
More informationVECTOR FUNCTIONS. which a space curve proceeds at any point.
3 VECTOR FUNCTIONS Tangent vectors show the direction in which a space curve proceeds at an point. The functions that we have been using so far have been real-valued functions. We now stud functions whose
More informationMath 233. Practice Problems Chapter 15. i j k
Math 233. Practice Problems hapter 15 1. ompute the curl and divergence of the vector field F given by F (4 cos(x 2 ) 2y)i + (4 sin(y 2 ) + 6x)j + (6x 2 y 6x + 4e 3z )k olution: The curl of F is computed
More informationSolutions for the Practice Final - Math 23B, 2016
olutions for the Practice Final - Math B, 6 a. True. The area of a surface is given by the expression d, and since we have a parametrization φ x, y x, y, f x, y with φ, this expands as d T x T y da xy
More informationSurface Integrals (6A)
Surface Integrals (6A) Surface Integral Stokes' Theorem Copright (c) 2012 Young W. Lim. Permission is granted to cop, distribute and/or modif this document under the terms of the GNU Free Documentation
More informationFinal Exam Review Sheet : Comments and Selected Solutions
MATH 55 Applied Honors alculus III Winter Final xam Review heet : omments and elected olutions Note: The final exam will cover % among topics in chain rule, linear approximation, maximum and minimum values,
More informationTopic 3 Notes Jeremy Orloff
Topic 3 Notes Jerem Orloff 3 Line integrals and auch s theorem 3.1 Introduction The basic theme here is that comple line integrals will mirror much of what we ve seen for multivariable calculus line integrals.
More informationAPJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE (SUPPLEMENTARY) EXAMINATION, FEBRUARY 2017 (2015 ADMISSION)
B116S (015 dmission) Pages: RegNo Name PJ BDUL KLM TECHNOLOGICL UNIVERSITY FIRST SEMESTER BTECH DEGREE (SUPPLEMENTRY) EXMINTION, FEBRURY 017 (015 DMISSION) MaMarks : 100 Course Code: M 101 Course Name:
More informationGreen s Theorem. Fundamental Theorem for Conservative Vector Fields
Assignment - Mathematics 4(Model Answer) onservative vector field and Green theorem onservative Vector Fields If F = φ, for some differentiable function φ in a domaind, then we say that F is conservative
More informationReview: critical point or equivalently f a,
Review: a b f f a b f a b critical point or equivalentl f a, b A point, is called a of if,, 0 A local ma or local min must be a critical point (but not conversel) 0 D iscriminant (or Hessian) f f D f f
More informationReview problems for the final exam Calculus III Fall 2003
Review problems for the final exam alculus III Fall 2003 1. Perform the operations indicated with F (t) = 2t ı 5 j + t 2 k, G(t) = (1 t) ı + 1 t k, H(t) = sin(t) ı + e t j a) F (t) G(t) b) F (t) [ H(t)
More informationMATHEMATICS 200 December 2011 Final Exam Solutions
MATHEMATICS December 11 Final Eam Solutions 1. Consider the function f(, ) e +4. (a) Draw a contour map of f, showing all tpes of level curves that occur. (b) Find the equation of the tangent plane to
More informationMath 67. Rumbos Fall Solutions to Review Problems for Final Exam. (a) Use the triangle inequality to derive the inequality
Math 67. umbos Fall 8 Solutions to eview Problems for Final Exam. In this problem, u and v denote vectors in n. (a) Use the triangle inequality to derive the inequality Solution: Write v u v u for all
More informationSection Arclength and Curvature. (1) Arclength, (2) Parameterizing Curves by Arclength, (3) Curvature, (4) Osculating and Normal Planes.
Section 10.3 Arclength and Curvature (1) Arclength, (2) Parameterizing Curves by Arclength, (3) Curvature, (4) Osculating and Normal Planes. MATH 127 (Section 10.3) Arclength and Curvature The University
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. D) D: (-, 0) (0, )
Midterm Practice Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the domain and graph the function. ) G(t) = t - 3 ) 3 - -3 - - 3 - - -3
More information39. (a) Use trigonometric substitution to verify that. 40. The parabola y 2x divides the disk into two
35. Prove the formula A r for the area of a sector of a circle with radius r and central angle. [Hint: Assume 0 and place the center of the circle at the origin so it has the equation. Then is the sum
More information18.02 Multivariable Calculus Fall 2007
MIT OpenourseWare http://ocw.mit.edu 8.02 Multivariable alculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 8.02 Lecture 8. hange of variables.
More informationReview Questions for Test 3 Hints and Answers
eview Questions for Test 3 Hints and Answers A. Some eview Questions on Vector Fields and Operations. A. (a) The sketch is left to the reader, but the vector field appears to swirl in a clockwise direction,
More informationVector Fields and Line Integrals The Fundamental Theorem for Line Integrals
Math 280 Calculus III Chapter 16 Sections: 16.1, 16.2 16.3 16.4 16.5 Topics: Vector Fields and Line Integrals The Fundamental Theorem for Line Integrals Green s Theorem Curl and Divergence Section 16.1
More information