Math Change of Variables in Triple Integrals

Transcription

1 Math Change of Variables in Triple Integrals Peter A. Perry University of Kentucky November 9, 2018

2 Homework e-rre-ead section 15.9 Finish work on section 15.9, problems 1-37 (odd) from tewart Begin reviewing for your exam, Wednesday, November 14

3 Unit III: Multiple Integrals Lecture 24 Lecture 25 Lecture 26 Lecture 27 Lecture 28 Lecture 29 Lecture 30 Lecture 31 Lecture 32 Lecture 33 Lecture 34 Double Integrals over ectangles Double Integrals over General egions Double Integrals in Polar Coodinates Applications of Double Integrals urface Area Triple Integrals Triple Integrals in Cylindrical Coordinates Triple Integrals in pherical Coordinates Change of Variable in Multiple Integrals, Part I Change of Variable in Multiple Integrals, Part II Exam III eview

4 Goals of the Day Understand what a transformation T between two regions in space is Understand how to compute the Jacobian Matrix and Jacobian determinant of a transformation and understand what the Jacobian determinant measures Understand how to compute triple integrals using the change of variables formula

5 Change of Variable: uv xy If x = g(u, v), y = h(u, v), and if the region in the uv plane is mapped to the region in the xy plane, then f (x, y) da = f (x(u, v), y(u, v)) (x, y) (u, v) du dv The Jacobian determinant (x, y) (u, v) = measures how areas change under the map (u, v) (x, y). We get the change of variables formula

6 Change of Variable: uvw to xyz If x = g(u, v, w), y = h(u, v, w), z = k(u, v, w) and the region in uvw space is mapped to in xyz space, then where f (x, y, z) dv = f (x(u, v, w), y(u, v, w), z(u, v, w)) (x, y, z) (u, v, w) du dv dw (x, y, z) (u, v, w) = z z w w z w

7 Cylindrical and pherical Coordinates ecall that the Jacobian determinant is (x, y, z) (u, v, w) = z z w w z w Find the Jacobian determinant if: (1) x = u cos v, y = u sin v, z = w (cylindrical) (2) x = u sin w cos v, y = u sin w sin v, z = u cos w (spherical) What s the connection with these formulas and formulas for integration in cylindrical and spherical coordinates?

8 v β Polar Coordinates α The transformation y a u = b b u x = u cos v, y = u sin v maps a rectangle in the uv plane to a polar rectangle in the xy plane The Jacobian of this transformation is cos v u sin v sin v u cos v = u u = a x

9 v β Polar Coordinates α The transformation y a u = b b u x = u cos v, y = u sin v maps a rectangle in the uv plane to a polar rectangle in the xy plane The Jacobian of this transformation is cos v u sin v sin v u cos v = u u = a x

10 v β Polar Coordinates α The transformation y a u = b b u x = u cos v, y = u sin v maps a rectangle in the uv plane to a polar rectangle in the xy plane The Jacobian of this transformation is cos v u sin v sin v u cos v = u u = a x

11 v β Polar Coordinates α The transformation y a u = b b u x = u cos v, y = u sin v maps a rectangle in the uv plane to a polar rectangle in the xy plane The Jacobian of this transformation is cos v u sin v sin v u cos v = u u = a x

12 v β Polar Coordinates α The transformation y a u = b b u x = u cos v, y = u sin v maps a rectangle in the uv plane to a polar rectangle in the xy plane The Jacobian of this transformation is cos v u sin v sin v u cos v = u u = a x

13 w Cylindrical Coordinates v The transformation x = u cos v, y = u sin v, z = w u maps a box in the uvw plane to a cylindrical wedge in xyz space z The Jacobian of this transformation is cos v u sin v 0 sin v u cos v = u y x

14 w pherical Coordinates The transformation u v x = u sin(w)cos(v) y = u sin(w) sin(v) z = u cos(w) z maps a box in the uvw plane to a spherical wedge in xyz space y The Jacobian of this transformation is u 2 sin(w) x

15 Volume of an Ellipsoid Find the volume enclosed by the ellipsoid using the transformation x 2 a 2 + y 2 b 2 + z2 c 2 = 1 x = au, y = bv, z = cw