W3V1 - Change of Variables PDF

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Change of Variables in Multiple Integrals In one-dimensional calculus, we often use substitution to re-write an integral: Z g(b) Z b ′ f (g(t))g (t)dt = f (x)dx a

g(a)

Graphically examine this example:

Z

3 2t

e dt = 0

Z

0

6

1 ex · dx 2

We can see this change of variable changes the function in the integrand and includes a scaling factor to make up for the change in the interval. We can use this single variable substitution inside of double integrals as well: Z 1Z 2 sin(πy ) dxdy 0 1 1 + 2x

Sometimes we want to substitute in a double integral using u = f (x, y) and v = g(x, y ) ZZ 3xdA Let D be the parallelogram with vertices (0, 0), (2, 4), (2, 1), (4, 5) and evaluate D

Let u = y − 2x and v = x − 2y, how can you write 3x and D in terms of u and v ?

Try

ZZ

3xdA in terms of u and v, do you get the same answer? D

Why were our answers different? Because we were missing the scaling factor! For any substitution x = f (u, v) and y = g(u, v), the scaling factor is called the Jacobian:      ∂x ∂x   ∂(x, y)   ∂u ∂v  ∂x ∂y ∂x ∂y    ∂(u, v)  =  ∂y ∂y  = ∂u ∂v − ∂v ∂u   ∂u ∂v   ZZ ZZ  ∂(x, y)    dA h(x, y)dA = This means that h(f (u, v), g(u, v))  ∂(u, v)  Duv Dxy    ∂(x, y)  ? If u = y − 2x and v = x − 2y, what is  ∂(u, v) 

This is exactly the scaling factor we needed!...