Change of Variables Jacobians Carl Gustav Jacobi Change

Change of Variables: Jacobians Carl Gustav Jacobi

Change of Variable - Single • In one-dimensional calculus, we often use a change of variable (a substitution) to simplify an integral. where x = g(u) and a = g(c), b = g(d).

How about Change of Double Variables ?

Definition – Jacobian • Let x = g(u, v) and y = h(u, v). Then the jacobian is given by:

Theorem

Jacobian

Rectangular to Polar coordinates x = g(r, θ) = r cos θ y = h(r, θ) = r sin θ Find the Jacobian.

Double Integral in Polar Coordinates

Your turn to show the triple Integral for Spherical Coordinates! i. e verify that dv = ρ2 sin Φ dρ dθ dΦ

Triple Integrals The Jacobian is 3 x 3 determinant:

Triple Integrals

TRIPLE INTEGRALS Recall that – The change of variables is given by: x = ρ sin Φ cos θ y = ρ sin Φ sin θ z = ρ cos Φ

We compute the Jacobian as follows:

• Since 0 ≤ Φ ≤ π , we have sin Φ ≥ 0. • Therefore,

• Thus, triple integral in spherical coordinates becomes:

Change of Variables in Multiple Integrals

Why do we change of Variables? 1. To get a simpler integrand 2. To transform the region into one that is much easier to deal with.

Example-1: Determine the new region if the ellipse is transformed by

Jacobian Transformation

Example-2: Evaluate Where R is the parallelogram with vertices (2, 0), (5, 3), (6, 7) and (3, 4) Using the transformation to R.

Home-work Use the change of variables x = u 2 – v 2, y = 2 uv to evaluate the integral where R is the region bounded by: . – The parabolas y 2 = 4 – 4 x and y 2 = 4 + 4 x, y ≥ 0.