Change of Variables & Jacobian

Jason Aran

June 3, 2015

Jason Aran Change of Variables & Jacobian June 3, 2015 1 / 20

Transformations - Definition

Definition

A Transformation T from the uv -plane to the xy -plane is a function thatmaps points in the uv -plane to points in the xy -plane by T (u, v) = (x , y).Notice that x = x(u, v) and y = y(u, v).

Jason Aran Change of Variables & Jacobian June 3, 2015 2 / 20

Additional Example From ClassSuppose in the rθ-plane you have the following region:

We showed in class that the transformation T : x = r cos θ, y = r sin θmaps the above region into the region shown below (in the xy -plane).

Jason Aran Change of Variables & Jacobian June 3, 2015 3 / 20

Transformations - One-to One Property

Definition

We say that T is One-to One if for every (x , y) in the image R, there isexactly one point (u, v) in the pre-image S such that T (u, v) = (x , y)

Jason Aran Change of Variables & Jacobian June 3, 2015 4 / 20

Example of a Transformation

Example

Consider the region R in the xy -plane which is enclosed by x − y = 0,x − y = 1, x + y = 1, and x + y = 3.

(a) Sketch R. Label all curves and their intersections.

(b) Using the transformation u = x − y and v = x + y to find thepre-image of R in the uv -plane. Sketch it, labelling all curves andtheir intersections.

(c) Find the inverse of the transformation; that is, solve for x and y interms of u and v .

Jason Aran Change of Variables & Jacobian June 3, 2015 5 / 20

Required Conditions

Suppose that we have a transformation T in which x = x(u, v) andy = y(u, v) satisfy the following conditions:

1 T is one-to-one.

2 x(u, v) and y(u, v) have continuous partial derivatives.

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The Main Idea

The area of a cross section in the xy -plane may not be exactly the same asthe area of a cross section in the uv -plane. We want to determine therelationship; that is, we want to determine the scaling factor that isneeded so that the areas are equal.

Jason Aran Change of Variables & Jacobian June 3, 2015 7 / 20

Image of S under T

Suppose that we start with a tiny rectangle as a cross section in theuv -plane with dimensions ∆u and ∆v . The image will be roughly aparallelogram (as long as our partition is small enough).

Jason Aran Change of Variables & Jacobian June 3, 2015 8 / 20

Image of S under T

Notice:

The image of A is A′; that is T (A) = A′.

The image of B is B ′; that is T (B) = B ′.

The image of C is C ′; that is T (C ) = C ′.

The image of D is D ′; that is T (D) = D ′.

Jason Aran Change of Variables & Jacobian June 3, 2015 9 / 20

Image of S under TWe can express the endpoints of S as A(u0, v0), B(u0 + ∆u),C (u0, v0 + ∆v), and D(u0 + ∆u, v0 + ∆v).

The endpoints of R will be

A′(x(u0, v0), y(u0, v0))

B ′(x(u0 + ∆u, v0), y(u0 + ∆u, v0))

C ′(x(u0, v0 + ∆v), y(u0, v0 + ∆v))

Jason Aran Change of Variables & Jacobian June 3, 2015 10 / 20

Area of R

Notice that R is a parallelogram with sides −→a and−→b . So, the area of R is∥∥∥−→a ×−→b ∥∥∥.

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Area of R

The components or −→a are:

−→a = 〈x(u0 + ∆u, v0)− x(u0, v0), y(u0 + ∆u, v0)− y(u0, v0), 0〉

≈⟨∂x∂u∆u, ∂y∂u∆u, 0

⟩≈ 〈xu∆u, yu∆u, 0〉

Jason Aran Change of Variables & Jacobian June 3, 2015 12 / 20

Area of R

Similarly, the components or−→b are:

−→b = 〈x(u0, v0 + ∆v)− x(u0, v0), y(u0, v0 + ∆v)− y(u0, v0), 0〉

≈⟨∂x∂v ∆v , ∂y∂v ∆v , 0

⟩≈ 〈xv∆v , yv∆v , 0〉

Jason Aran Change of Variables & Jacobian June 3, 2015 13 / 20

Area of R

First, notice that the cross product of −→a and−→b is:

−→a ×−→b =

∣∣∣∣∣∣−→i

−→j

−→k

xu∆u yu∆u 0xv∆v yv∆v 0

∣∣∣∣∣∣= [(xu∆u)(yv∆v)− (xv∆v)(yu∆u)]

−→k

= (xuyv − xvyu)∆u∆v−→k

And, the area of R is:∥∥∥−→a ×−→b ∥∥∥ = |xuyv − xvyu|∆u∆v

That is,∆AR = |xuyv − xvyu|∆u∆v

Jason Aran Change of Variables & Jacobian June 3, 2015 14 / 20

The Jacobian Matrix

What we have just shown is that the area of a cross section of region R is:

∆AR = |xuyv − xvyu|∆u∆v

And, the area of a cross section of region S is:

∆AS = ∆u∆v

So, the the scaling factor that relates the two is |xuyv − xvyu|. We oftenwrite this as the determinant of a matrix, called the Jacobian Matrix.

Definition

The Jacobian Matrix is ∂(x ,y)∂(u,v) =

(xu xvyu yv

).

Jason Aran Change of Variables & Jacobian June 3, 2015 15 / 20

Jacobian Examples

Example

Calculate the Jacobian (the determinant of the Jacobian Matrix) for thefollowing transformations:

1 Polar: x = r cos θ, y = r sin θ

2 Cylindrical: x = r cos θ, y = r sin θ, z = z

3 Spherical: x = ρ cos θ sinφ, y = ρ sin θ sinφ, z = ρ cosφ

Example

Calculate the Jacobian for the transformation described in slide 4:x = 1

2(u + v), y = 12(v − u)

Jason Aran Change of Variables & Jacobian June 3, 2015 16 / 20

Theorem

Theorem

Suppose that the region S in the uv-plane is mapped onto the region R inthe xy-plane by the one-to-one transformation T defined by x = x(u, v),y = y(u, v). where x and y have continuous first order partial derivatives

on S. If f is continuous on R and the Jacobian∣∣∣∂(x ,y)∂(u,v)

∣∣∣ is non-zero on S,

then ∫∫R

f (x , y) dAxy =

∫∫S

f (x(u, v), y(u, v))

∣∣∣∣∂(x , y)

∂(u, v)

∣∣∣∣ dAuv

Jason Aran Change of Variables & Jacobian June 3, 2015 17 / 20

Comments on the theorem

Notice that when we apply the theorem, we convert

1 The function

2 The differential

3 The limits of integration

We typically apply the theorem if

1 The region over which we are integrating is particularly difficult

2 The function we are trying to integrate is particularly difficult

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Example 1

Example

Evaluate∫∫

Rx−yx+y dA where R is the region enclosed by x − y = 0,

x − y = 1, x + y = 1, and x + y = 3.

Jason Aran Change of Variables & Jacobian June 3, 2015 19 / 20

Example 2

Example

Find the volume of the solid enclosed by the ellipsoid x2

4 + y2

25 + z2

9 = 1 bytransforming the ellipsoid into a sphere of radius 1.Hint: Use the transformation x = 2u, y = 5v , z = 3w .

Jason Aran Change of Variables & Jacobian June 3, 2015 20 / 20