MATH 19520/51 Class 15

Transcription

1 MATH 19520/51 Class 15 Minh-Tam Trinh University of Chicago

2 1 Change of variables in two dimensions. 2 Double integrals via change of variables.

3 Change of Variables Slogan: An n-variable substitution = a geometric transformation of n-dimensional space.

4 Change of Variables Slogan: An n-variable substitution = a geometric transformation of n-dimensional space. What do we mean by geometric transformation?

5 n = 1: A substitution that expresses x in terms of u is a transformation from the u-axis to the x-axis. What are the following transformations?

6 n = 1: A substitution that expresses x in terms of u is a transformation from the u-axis to the x-axis. What are the following transformations? 1 x = 2u. 2 x = u. 3 x = u x = u. 5 x = u x = (u + 1).

7 n = 1: A substitution that expresses x in terms of u is a transformation from the u-axis to the x-axis. What are the following transformations? 1 x = 2u. Scale to 2 times as large 2 x = u. 3 x = u x = u. 5 x = u x = (u + 1).

8 n = 1: A substitution that expresses x in terms of u is a transformation from the u-axis to the x-axis. What are the following transformations? 1 x = 2u. Scale to 2 times as large 2 x = u. Flip direction 3 x = u x = u. 5 x = u x = (u + 1).

9 n = 1: A substitution that expresses x in terms of u is a transformation from the u-axis to the x-axis. What are the following transformations? 1 x = 2u. Scale to 2 times as large 2 x = u. Flip direction 3 x = u + 1. Shift right by 1 unit 4 x = u. 5 x = u x = (u + 1).

10 n = 1: A substitution that expresses x in terms of u is a transformation from the u-axis to the x-axis. What are the following transformations? 1 x = 2u. Scale to 2 times as large 2 x = u. Flip direction 3 x = u + 1. Shift right by 1 unit 4 x = u. Keep positive axis in place, flip negative axis onto it 5 x = u x = (u + 1).

11 n = 1: A substitution that expresses x in terms of u is a transformation from the u-axis to the x-axis. What are the following transformations? 1 x = 2u. Scale to 2 times as large 2 x = u. Flip direction 3 x = u + 1. Shift right by 1 unit 4 x = u. Keep positive axis in place, flip negative axis onto it 5 x = u + 1. First flip direction, then shift right by 1 6 x = (u + 1).

12 n = 1: A substitution that expresses x in terms of u is a transformation from the u-axis to the x-axis. What are the following transformations? 1 x = 2u. Scale to 2 times as large 2 x = u. Flip direction 3 x = u + 1. Shift right by 1 unit 4 x = u. Keep positive axis in place, flip negative axis onto it 5 x = u + 1. First flip direction, then shift right by 1 6 x = (u + 1). First shift right by 1, then flip direction

13 n = 2: A substitution that turns x and y into u and v is a transformation from the uv-plane to the xy-plane: Formulas for x and y in terms of u and v let us transform S (in the uv-plane) into R (in the xy-plane).

14 What are the following transformations?

15 What are the following transformations? 1 (x, y) = (u + 1, v). 2 (x, y) = (u 1, v + 1). 3 (x, y) = (v, u). 4 (x, y) = (u, u + v). 5 (x, y) = (u, 0). 6 (x, y) = (u cos v, u sin v). Also, which of these substitutions can you undo?

16 What are the following transformations? 1 (x, y) = (u + 1, v). Shift right by 1 2 (x, y) = (u 1, v + 1). 3 (x, y) = (v, u). 4 (x, y) = (u, u + v). 5 (x, y) = (u, 0). 6 (x, y) = (u cos v, u sin v). Also, which of these substitutions can you undo?

17 What are the following transformations? 1 (x, y) = (u + 1, v). Shift right by 1 2 (x, y) = (u 1, v + 1). Shift left by 1 and up by 1 3 (x, y) = (v, u). 4 (x, y) = (u, u + v). 5 (x, y) = (u, 0). 6 (x, y) = (u cos v, u sin v). Also, which of these substitutions can you undo?

18 What are the following transformations? 1 (x, y) = (u + 1, v). Shift right by 1 2 (x, y) = (u 1, v + 1). Shift left by 1 and up by 1 3 (x, y) = (v, u). Flip diagonally, interchanging right and up 4 (x, y) = (u, u + v). 5 (x, y) = (u, 0). 6 (x, y) = (u cos v, u sin v). Also, which of these substitutions can you undo?

19 What are the following transformations? 1 (x, y) = (u + 1, v). Shift right by 1 2 (x, y) = (u 1, v + 1). Shift left by 1 and up by 1 3 (x, y) = (v, u). Flip diagonally, interchanging right and up 4 (x, y) = (u, u + v). Shear vertically by 1 unit 5 (x, y) = (u, 0). 6 (x, y) = (u cos v, u sin v). Also, which of these substitutions can you undo?

20 What are the following transformations? 1 (x, y) = (u + 1, v). Shift right by 1 2 (x, y) = (u 1, v + 1). Shift left by 1 and up by 1 3 (x, y) = (v, u). Flip diagonally, interchanging right and up 4 (x, y) = (u, u + v). Shear vertically by 1 unit 5 (x, y) = (u, 0). Collapse the vertical direction 6 (x, y) = (u cos v, u sin v). Also, which of these substitutions can you undo?

21 What are the following transformations? 1 (x, y) = (u + 1, v). Shift right by 1 2 (x, y) = (u 1, v + 1). Shift left by 1 and up by 1 3 (x, y) = (v, u). Flip diagonally, interchanging right and up 4 (x, y) = (u, u + v). Shear vertically by 1 unit 5 (x, y) = (u, 0). Collapse the vertical direction 6 (x, y) = (u cos v, u sin v). Turn rectangles into polar wedges Also, which of these substitutions can you undo?

22 A substitution is called one-to-one, or invertible, iff you can undo it.

23 A substitution is called one-to-one, or invertible, iff you can undo it. The polar-coordinate substitution is invertible as long as r 0.

24 The substitution x = x(u, v) and y = y(u, v): Transforms the uv-plane into the xy-plane. Pulls back functions of x and y to functions of u and v. (Reverse direction!)

25 The substitution x = x(u, v) and y = y(u, v): Transforms the uv-plane into the xy-plane. Pulls back functions of x and y to functions of u and v. (Reverse direction!) What do the following do to the function f (x, y) = x 2 y 2? 1 (x, y) = (u, v). 2 (x, y) = (v, u). 3 (x, y) = ( u, v). 4 (x, y) = (u + 1, v).

26 The substitution x = x(u, v) and y = y(u, v): Transforms the uv-plane into the xy-plane. Pulls back functions of x and y to functions of u and v. (Reverse direction!) What do the following do to the function f (x, y) = x 2 y 2? 1 (x, y) = (u, v). u 2 v 2 2 (x, y) = (v, u). 3 (x, y) = ( u, v). 4 (x, y) = (u + 1, v).

27 The substitution x = x(u, v) and y = y(u, v): Transforms the uv-plane into the xy-plane. Pulls back functions of x and y to functions of u and v. (Reverse direction!) What do the following do to the function f (x, y) = x 2 y 2? 1 (x, y) = (u, v). u 2 v 2 2 (x, y) = (v, u). v 2 u 2 (flipped on a diagonal) 3 (x, y) = ( u, v). 4 (x, y) = (u + 1, v).

28 The substitution x = x(u, v) and y = y(u, v): Transforms the uv-plane into the xy-plane. Pulls back functions of x and y to functions of u and v. (Reverse direction!) What do the following do to the function f (x, y) = x 2 y 2? 1 (x, y) = (u, v). u 2 v 2 2 (x, y) = (v, u). v 2 u 2 (flipped on a diagonal) 3 (x, y) = ( u, v). u 2 v 2 (no change) 4 (x, y) = (u + 1, v).

29 The substitution x = x(u, v) and y = y(u, v): Transforms the uv-plane into the xy-plane. Pulls back functions of x and y to functions of u and v. (Reverse direction!) What do the following do to the function f (x, y) = x 2 y 2? 1 (x, y) = (u, v). u 2 v 2 2 (x, y) = (v, u). v 2 u 2 (flipped on a diagonal) 3 (x, y) = ( u, v). u 2 v 2 (no change) 4 (x, y) = (u + 1, v). (u + 1) 2 v 2 (shift left in the first coordinate)

30 Why we care about this geometric interpretation: We solve lots of integrals using variable substitution, e.g., (1) or (2) b a cos(2x + 1) dx = e (x2 +y 2) dx dy = 2b+1 2a+1 2π 0 1 cos(u) du 2 0 e u2 u du dv.

31 Why we care about this geometric interpretation: We solve lots of integrals using variable substitution, e.g., (1) or b a cos(2x + 1) dx = 2b+1 2a+1 1 cos(u) du 2 (2) What s really going on: e (x2 +y 2) dx dy = 2π 0 0 e u2 u du dv. We re pulling back functions from xy-coordinates to uv-coordinates, then integrating the new function in uv-coordinates.

32 Key point: A substitution that expresses x in terms of u doesn t have to preserve length. In fact, in turns (3) dx into dx du du. Above, dx du keeps track of the distortion in length.

33 Key point: A substitution that expresses x in terms of u doesn t have to preserve length. In fact, in turns (3) dx into dx du du. Above, dx du Example To solve b a keeps track of the distortion in length. 2 cos(2x + 1) dx, we take u = 2x + 1.

34 Key point: A substitution that expresses x in terms of u doesn t have to preserve length. In fact, in turns (3) dx into dx du du. Above, dx du Example To solve b a keeps track of the distortion in length. 2 cos(2x + 1) dx, we take u = 2x + 1. This is the same as taking x = u 1 2, which gives (4) dx du = 1 2. That s why we get b a cos(2x + 1) dx = 2b+1 2a+1 cos(u) ( 1 2 du ).

35 Example (Stewart, 15.9, Example 1) An example of distortion in area: This is what happens to the unit square {(u, v) : 0 u 1 and 0 v 1} under the substitution (x, y) = (u 2 v 2, 2uv).

36 The Jacobian In single-variable calculus, when you do a u-substitution, we turn (5) dx = stuff(u) du and we know stuff(u) = dx du.

37 The Jacobian In single-variable calculus, when you do a u-substitution, we turn (5) dx = stuff(u) du and we know stuff(u) = dx du. In 2-variable calculus, (6) dx dy = stuff(u, v) du dv. But what is stuff(u, v)?

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39 Answer: The Jacobian of a substitution from (x, y) into (u, v) is (7) x u y u x v y v Stewart denotes the Jacobian by = x uy v x v y u. (x, y) (u, v).

40 Double Integrals via Change of Variables The change-of-variables formula in 2 dimensions: If a substitution from (x, y) into (u, v) transforms S into R and is one-to-one (except possibly at the boundary of S), then: (8) R f (x, y) dx dy = S f (x(u, v), y(u, v)) (x, y) du dv. (u, v)

41 Example The polar integral is a special case of the change-of-variables formula: What is the Jacobian of the substitution (x, y) = (r cos θ, r sin θ)?

42 Example The polar integral is a special case of the change-of-variables formula: What is the Jacobian of the substitution (x, y) = (r cos θ, r sin θ)? (9) x r y θ x θ y r = (cos θ)(r cos θ) ( r sin θ)(sin θ) = r cos 2 θ + r sin 2 θ = r.

43 Example The polar integral is a special case of the change-of-variables formula: What is the Jacobian of the substitution (x, y) = (r cos θ, r sin θ)? (9) x r y θ x θ y r = (cos θ)(r cos θ) ( r sin θ)(sin θ) = r cos 2 θ + r sin 2 θ = r. This is why the polar change-of-variables formula involves turning (10) dx dy into r dr dθ.