Multiple Integrals: Change of Coordinates
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1 Multiple Integrals: Change of Coordinates Calculus III Josh Engwer TTU 9 October 04 Josh Engwer (TTU) Multiple Integrals: Change of Coordinates 9 October 04 / 8
2 Change of Coordinates in Two Variables Proposition Let D R be a closed & bounded region in the xy-plane. Let D R be a closed & bounded region in the uv-plane. Let f C(D). Let transformation T map region D to region D s.t. T : { x = T (u, v) y = T (u, v) where T, T C (,) (D ) Then: where D := det (u, v) f (x, y) da = f [T (u, v), T (u, v)] D (u, v) da [ ] u v is called the Jacobian. u v PROOF: Take Linear Algebra & Advanced Calculus. Josh Engwer (TTU) Multiple Integrals: Change of Coordinates 9 October 04 / 8
3 The Jacobian (Rectangular Polar Coordinates) Let transformation T map from Rectangular Polar T : Then: (r, θ) [ = det r r θ θ [ cos θ r sin θ = det sin θ r cos θ ] ] { x = r cos θ y = r sin θ D = r( cos θ + r sin θ) = r cos θ + sin θ = r f (x, y) da = f (r cos θ, r sin θ) D (r, θ) = f (r cos θ, r sin θ) r dr dθ D dr dθ Josh Engwer (TTU) Multiple Integrals: Change of Coordinates 9 October 04 3 / 8
4 Change of Coordinates in Two Variables Sometimes it s easier to write a transformation in terms of x, y instead of u, v like so: { u = T T : (x, y) v = T (x, y) where T, T C (,) (D) In other words, sometimes it s easier to work with an inverse transformation (instead of a transformation.) Then, to compute the Jacobian (u, v) : [ ] (u, v) u u Compute Jacobian := det Solve equation v v (u, v) = for (u, v) (u, v) Josh Engwer (TTU) Multiple Integrals: Change of Coordinates 9 October 04 4 / 8
5 Transformation T may Simplify Region of Integration Notice that transformation T maps from the uv-plane to the xy-plane (not the other way around as one would have expected.) Josh Engwer (TTU) Multiple Integrals: Change of Coordinates 9 October 04 5 / 8
6 Transformation T may Simplify Region of Integration Notice that transformation T maps from the uv-plane to the xy-plane (not the other way around as one would have expected.) Josh Engwer (TTU) Multiple Integrals: Change of Coordinates 9 October 04 6 / 8
7 Transformation T may Simplify Region of Integration Notice that transformation T maps from the uv-plane to the xy-plane (not the other way around as one would have expected.) Josh Engwer (TTU) Multiple Integrals: Change of Coordinates 9 October 04 7 / 8
8 Inverse Transformation T may Simplify the Integrand Let transformation T : Jacobian (u, v) Now, D (u, v) = det { u = x 3y v = x + 3y. [ u v u v ] Then: [ 3 = det 3 ] = ()(3) ( 3)() = = = () = = (u, v) (u, v) (u, v) = x 3y x + 3y da CV u = D v (u, v) da = D u v da Josh Engwer (TTU) Multiple Integrals: Change of Coordinates 9 October 04 8 / 8
9 Change of Coordinates in Three Variables PROOF: Take Linear Algebra & Advanced Calculus. Josh Engwer (TTU) Multiple Integrals: Change of Coordinates 9 October 04 9 / 8 Proposition Let E R 3 be a closed & bounded solid in xyz-space. Let E R 3 be a closed & bounded solid in uvw-space. Let f C(E). Let transformation T map solid E to solid E s.t. x = T (u, v, w) T : y = T (u, v, w) where T, T, T 3 C (,,) (E ) z = T 3 (u, v, w) Then: where f (x, y, z) dv = f [T (u, v, w), T (u, v, w), T 3 (u, v, w)] (x, y, z) E (u, v, w) dv (x, y, z) u v w := det is called the Jacobian. (u, v, w) u v w E u v w
10 The Jacobian (Rectangular Cylindrical Coordinates) x = r cos θ Let T map from Rectangular Cylindrical T : y = r sin θ z = z Then: (x, y, z) (r, θ, z) = det = det r r r θ θ θ cos θ r sin θ 0 sin θ r cos θ = cos θ r cos θ 0 0 ( r sin θ) sin θ = r cos θ + r sin θ ( ) = r cos θ + sin θ = r Josh Engwer (TTU) Multiple Integrals: Change of Coordinates 9 October 04 0 / 8
11 The Jacobian (Rectangular Spherical Coordinates) x = ρ sin φ cos θ Let T map from Rectangular Spherical T : y = ρ sin φ sin θ z = ρ cos φ (x, y, z) (ρ, θ, φ) = det = det = ρ ρ ρ θ θ θ φ φ φ sin φ cos θ ρ sin φ sin θ ρ cos φ cos θ sin φ sin θ ρ sin φ cos θ ρ cos φ sin θ cos φ 0 ρ sin φ ρ sin 3 φ cos θ ρ sin 3 φ sin θ ρ cos φ sin φ sin θ ρ cos φ sin φ cos θ = ρ sin 3 φ [ cos θ + sin θ ] ρ cos φ sin φ [ sin θ + cos θ ] = ρ sin 3 φ ρ cos φ sin φ = ρ sin φ [ sin φ + cos φ ] = ρ sin φ Josh Engwer (TTU) Multiple Integrals: Change of Coordinates 9 October 04 / 8
12 Change of Coordinates in Three Variables Sometimes it s easier to write a transformation in terms of x, y, z instead of u, v, w like so: u = T T (x, y, z) : v = T (x, y, z) where T w = T, T, T 3 C (,,) (E) 3 (x, y, z) Then, to compute the Jacobian Compute Jacobian Solve equation (u, v, w) (x, y, z) (x, y, z) (u, v, w) : := det u v w u v w u v w (x, y, z) (u, v, w) (x, y, z) = for (u, v, w) (x, y, z) (u, v, w) Josh Engwer (TTU) Multiple Integrals: Change of Coordinates 9 October 04 / 8
13 Transformation T may Simplify Solid of Integration Notice that transformation T maps from uvw-space to xyz-space (not the other way around as one would have expected.) Josh Engwer (TTU) Multiple Integrals: Change of Coordinates 9 October 04 3 / 8
14 Linear Transformations in R Definition (Linear Transformation in R ) { x = au + bv Transformation T is linear if T :, where a, b, c, d R y = cu + dv { u = ax + by Inverse transformation T is linear if T :, with a, b, c, d R v = cx + dy Theorem Transformation T is linear inverse transformation T is linear. Typical examples of linear transformations: Dilations Rotations Relections Shears NOTE: This is an overview of linear transformations take Linear Algebra. Josh Engwer (TTU) Multiple Integrals: Change of Coordinates 9 October 04 4 / 8
15 Linear Transformations in R 3 Definition (Linear Transformation in R 3 ) x = a u + a v + a 3 w Transformation T is linear if T : y = a u + a v + a 3 w, with a,..., a 33 R z = a 3 u + a 3 v + a 33 w u = a x + a y + a 3 z Inverse trans. T is linear if T : v = a x + a y + a 3 z, a,..., a 33 R w = a 3 x + a 3 y + a 33 z Theorem Transformation T is linear inverse transformation T is linear. Typical examples of linear transformations: Dilations Rotations Relections Shears NOTE: This is an overview of linear transformations take Linear Algebra. Josh Engwer (TTU) Multiple Integrals: Change of Coordinates 9 October 04 5 / 8
16 Double Integrals (Coordinate Change Procedure) SETUP: Let transformation T map region D to region D s.t. T : { x = T (u, v) y = T (u, v) where T, T C (,) (D ) Sketch region D in xy-plane & label BC s Apply transformation to each BC of region D to obtain a BC of region D 3 Sketch region D in uv-plane & label BC s & BP s 4 Compute the Jacobian in terms of u & v. (u, v) 5 Compute the absolute value of Jacobian (u, v) 6 f (x, y) da = f [T (u, v), T (u, v)] D D (u, v) da Josh Engwer (TTU) Multiple Integrals: Change of Coordinates 9 October 04 6 / 8
17 Double Integrals (Coordinate Change Procedure) SETUP: Let inverse transformation T map region D to region D s.t. { u = T T : (x, y) v = T (x, y) where T, T C (,) (D) Sketch region D in xy-plane & label BC s Apply transformation to each BC of region D to obtain a BC of region D Trivial if BC s of D are of the form: T (x, y) = k, T (x, y) = k, T (x, y) = k 3,..., where k, k, k 3, R If T is linear, solve linear system to obtain transformation T in terms of u, v. 3 Sketch region D in uv-plane & label BC s & BP s 4 Compute Jacobian in terms of u & v by solving (u, v) (u, v) 5 Compute the absolute value of Jacobian (u, v) 6 f (x, y) da = f [T (u, v), T (u, v)] D D (u, v) da (u, v) = Josh Engwer (TTU) Multiple Integrals: Change of Coordinates 9 October 04 7 / 8
18 Fin Fin. Josh Engwer (TTU) Multiple Integrals: Change of Coordinates 9 October 04 8 / 8
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v 1 v 3 u v = (( 1)4 (3)2, [1(4) ( 2)2], 1(3) ( 2)( 1)) = ( 10, 8, 1) (d) u (v w) = (u w)v (u v)w (Relationship between dot and cross product)
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