Mathematical Notation Math Calculus & Analytic Geometry III

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1 Name : Mathematical Notation Math alculus & Analytic Geometry III Use Word or WordPerect to recreate the ollowing documents. Each article is worth 10 points and can e printed and given to the instructor or ed to the instructor at james@richland.edu. Type your name at the top o each page. For expressions or equations, you should use the equation editor in Word or WordPerect. The documents were created using a 14 pt Times New oman ont with 0.75 margins. The equations were created using 14 pt as the standard ont size to make things readale. For individual symols (θ, α, etc), you can insert symols. In Word, use Insert / Symol and choose the Symol ont. For WordPerect, use trl-w and choose the Greek set. The due date or each o these documents is the day o the exam or that chapter. Late work will e accepted ut will lose 10% per class period.

2 hapter 12 - Vectors Dot Products The dot product is a scalar and is deined as uv 1 1+ uv 2 2+ uv 3 3 u v The dot product is 0 i and only i the vectors are orthogonal. u v u v cosθ v proj v W F cosθ PQ F PQ The orthogonal projection o v onto is 2 Work can e ound y ross Products i j k u v u u u v v v The cross product is a vector and is deined as The cross product is orthogonal to oth vectors. u v u v sinθ The area o a parallelogram is equal to the cross product o the adjacent side vectors. The cross product is the zero vector i and only i the vectors are parallel. Triple Scalar Products The triple scalar product is a scalar and is deined as u ( v w) u1 u2 u3 v1 v2 v3 w w w The asolute value o the triple scalar product is the volume o a parallelpiped. The triple scalar product is 0 i and only i the vectors are coplanar.

3 hapter 13 - Vector Valued Functions The derivatives o dot and cross products ollow the product rule. ( r1 r2) r 1 r2 + r1 r 2 and ( r1 r2) r 1 r2 + r1 r2 I a vector valued unction has constant length, the r and r' are orthogonal. The arc length o a smooth vector valued unction is The chain rule is d r d r dt dτ dt dτ The arc length parametrization is For a smooth vector valued unction, t dr s du t0 du dr ds dt dt and L a dr dt dr 1 ds To ind the unit tangent vector, take the derivative and normalize it. T() t dt r r To ind the unit normal vector, take the derivative o the unit tangent vector and normalize it. N() t T T For curves parametrized y arc length, T s r s and N( s) r r The inormal vector is the cross product o the unit tangent and unit normal vectors. It is also a unit vector and is ound y. The curvature is deined y ( t) ( t) ( t) B T N dt κ ds ( s) r s or κ The radius o the oscillating circle is called the radius o curvature and is T r ( s) ( s) 1 ρ κ ( t)

4 hapter 14 - Partial Derivatives The irst-order partial derivative o with respect to x is denoted y x ( x, is ound y inding the derivative o with every variale other than x treated as a constant. The second-order partial derivatives o are, yx y x 2 2 xx, yy 2 x and and the mixed partials are. I is continuous, the. Notice xy the ordering on the partials. The order or The order or 2 x y x y xy is right to let, y irst and x second. is rom let to right, x irst and y second. I z ( x, is dierentiale at ( x0, y 0), then the total dierential o at ( x0, y0) is dz ( x, y ) dx + ( x, y ) dy x 0 0 y 0 0 The chain rule says that i dz dx dy + dt dt dt z z z + u u u z ( x, and x and y are oth unctions o t, then. Furthermore, i x and y are oth unctions o u and v, then and z z z + v v v xy yx The gradient o is a vector deined y x, y, z xi + yj + zk x y +, xi y. is read "del ". j or The directional derivative o in the direction o the unit vector u can e written as (, ) (, ) D x y x y u u The applications o the gradient are too numerous to it on this page. It will appear in many ormulas.

5 hapter 15 - Multiple Integrals To evaluate a deinite integral, work rom inside to outside. The order o the integration is important, so e sure to use proper notation. g2( x) d h2( 1 1 x, y da x, y dy dx x, y dx dy a g x c h y I you can split the integrand into two independent unctions o x and y and the limits are constant, then you can split a doule (or triple) integral up into the product o the d d integrals. x g y dy dx x dx g y dy a c c a The area o a region is A da. The volume o a solid G is The surace area o a parametrically deined surace σ is The center o gravity ( x, o a lamina is given y M 1 x x x y da M mass o S V G r r u v y x δ (, ), δ (, ) dv da M 1 y y x y da M mass o The Theorem o Pappus says that the volume o a solid y revolving a region aout a line L is the area o the region times the distance traveled y the centroid. For polar and cylindrical coordinates, you need to insert an extra r into the integrand. For spherical coordinates, you need to insert an extra ρ 2 sinφ into the integrand. All o this is related to the Jacoian. I T is a transormation rom the uv plane into the xy plane, then the Jacoian o T is denoted y (, ) J u v ( xy, ) u u ( uv, ) x y v v. When applying the transormation, multiply the integrand y the asolute value o the Jacoian.

6 hapter 16 - Topics in Vector alculus ( x, y, z) ( x, y, z) + g ( x, y, z) + h( x, y, z) F i j k g h F + + F z onsider the vector unction. The divergence o F is a scalar deined y div The curl o F is a vector deined y curl i j k F F z g h The line integral o with respect to s along is the net signed area etween the curve and the graph o and is denoted y. ( x, y ) (, ) L Arc length can e expressed as. ds A x y ds The value o a line integral does not depend on its parametrization. However, i the orientation is reversed, the sign o the integral with respect to x and y changes, ut the integral with respect to the arc length parameter s remains unchanged. The work perormed y the vector ield is I F is a conservative vector ield and theorem o calculus applies to line integrals and F ( x, d r φ( x, y ) φ( x, y ) F W F T ds F d r ( x, φ( x, then the irst undamental I F is a conservative vector ield, then F d r 0 curve and the integral is independent o the path. or every piecewise smooth closed A vector ield in 2-space is conservative i Green's Theorem says (, ) (, ) g and in 3-space i g x y dx + g x y dy da curl F 0