Changes

=The Jacobian=

=The Jacobian=

Much of what happens in multivariable calculus involves the ''[[partial derivative]]s'' of the coordinates in one system with respect to the coordinates in the other:<math>\frac{\partial x}{\partial \theta}</math> for example.  It is very useful to represent these as a matrix:

Much of what happens in multivariable calculus involves the ''[[partial derivative]]s'' of the coordinates in one system with respect to the coordinates in the other.  It is very useful to represent these as a matrix:

::<math>\begin{pmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta}\end{pmatrix}\,</math>

::<math>\begin{bmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta}\end{bmatrix}\,</math>

Or, going in the other direction:

Or, going in the other direction:

::<math>\begin{pmatrix} \frac{\partial r}{\partial x} & \frac{\partial r}{\partial y} \\ \frac{\partial \theta}{\partial x} & \frac{\partial \theta}{\partial y}\end{pmatrix}\,</math>

::<math>\begin{bmatrix} \frac{\partial r}{\partial x} & \frac{\partial r}{\partial y} \\ \frac{\partial \theta}{\partial x} & \frac{\partial \theta}{\partial y}\end{bmatrix}\,</math>

This is the ''derivative matrix'' of the transformation from one coordinate system to the other.  It is often called the '''Jacobian matrix''', though some authors use the term "Jacobian" specifically to refer to the matrix's determinant.  This point is not important&mdash;it will not lead to confusion.

This is the ''derivative matrix'' of the transformation from one coordinate system to the other.  It is often called the '''Jacobian matrix''', though some authors use the term "Jacobian" specifically to refer to the matrix's determinant.  To avoid any confusion on this point, we will always put square brackets around matrices, and use absolute value signs to denote determinants.

The notation for the Jacobian matrix of u/v/w in terms of x/y/z is written:

The notation for the Jacobian matrix of u/v/w in terms of x/y/z is written:

::<math>J(u, v, w / x, y, z) = \frac{\partial(u, v, w)}{\partial(x, y, z)} = \begin{pmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} & \frac{\partial u}{\partial z} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z} \\ \frac{\partial w}{\partial x} & \frac{\partial w}{\partial y} & \frac{\partial w}{\partial z}\end{pmatrix} </math>

::<math>J(u, v, w / x, y, z) = \frac{\partial(u, v, w)}{\partial(x, y, z)} = \begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} & \frac{\partial u}{\partial z} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z} \\ \frac{\partial w}{\partial x} & \frac{\partial w}{\partial y} & \frac{\partial w}{\partial z}\end{bmatrix} </math>

More generally, it is:

More generally, it is:

::<math>\frac{\partial(f_1,f_2,...,f_n)}{\partial(x_1,x_2,...,x_n)} = \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & ... & \frac{\partial f_1}{\partial x_n} \\ ... &  & ... \\ \frac{\partial f_3}{\partial x_1} &...&\frac{\partial f_n}{\partial x_n}\end{pmatrix} </math>

::<math>\frac{\partial(f_1,f_2,...,f_n)}{\partial(x_1,x_2,...,x_n)} = \begin{bmatrix} \frac{\partial f_1}{\partial x_1} & ... & \frac{\partial f_1}{\partial x_n} \\ ... &  & ... \\ \frac{\partial f_3}{\partial x_1} &...&\frac{\partial f_n}{\partial x_n}\end{bmatrix} </math>

and the determinant is:

and the determinant is:

::<math>\begin{vmatrix} \frac{\partial f_1}{\partial x_1} & ... & \frac{\partial f_1}{\partial x_n} \\ ... &  & ... \\ \frac{\partial f_3}{\partial x_1} &...&\frac{\partial f_n}{\partial x_n}\end{vmatrix} </math>

::<math>\begin{vmatrix} \frac{\partial f_1}{\partial x_1} & ... & \frac{\partial f_1}{\partial x_n} \\ ... &  & ... \\ \frac{\partial f_3}{\partial x_1} &...&\frac{\partial f_n}{\partial x_n}\end{vmatrix} </math>

For polar coordinates, the Jacobian matrices are:

For polar coordinates, the Jacobian matrices are:

::<math>J(r, \theta / x, y) = \frac{\partial(r, \theta)}{\partial(x, y)} = \begin{pmatrix} \frac{\partial r}{\partial x} & \frac{\partial r}{\partial y} \\ \frac{\partial \theta}{\partial x} & \frac{\partial \theta}{\partial y} \end{pmatrix}</math>

::<math>J(r, \theta / x, y) = \frac{\partial(r, \theta)}{\partial(x, y)} = \begin{bmatrix} \frac{\partial r}{\partial x} & \frac{\partial r}{\partial y} \\ \frac{\partial \theta}{\partial x} & \frac{\partial \theta}{\partial y} \end{bmatrix}</math>

and:

and:

::<math>J(x, y / r, \theta) = \frac{\partial(x, y)}{\partial(r, \theta)} = \begin{pmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \end{pmatrix}</math>

::<math>J(x, y / r, \theta) = \frac{\partial(x, y)}{\partial(r, \theta)} = \begin{bmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \end{bmatrix}</math>

Working these out, we have

Working these out, we have

::<math>J(x, y / r, \theta) = \begin{pmatrix} \cos \theta & - r \sin \theta \\ \sin \theta & r \cos \theta \end{pmatrix}</math>

::<math>J(x, y / r, \theta) = \begin{bmatrix} \cos \theta & - r \sin \theta \\ \sin \theta & r \cos \theta \end{bmatrix}</math>

with determinant r, and:

with determinant r, and:

::<math>J(r, \theta / x, y) = \begin{pmatrix} \frac{x}{\sqrt{x^2 + y^2}} & \frac{y}{\sqrt{x^2 + y^2}} \\ \frac{- y}{x^2 + y^2} & \frac{x}{x^2 + y^2} \end{pmatrix} = \begin{pmatrix} \cos \theta & \sin \theta \\ \frac{- \sin \theta}{r} & \frac{\cos \theta}{r} \end{pmatrix}</math>

::<math>J(r, \theta / x, y) = \begin{bmatrix} \frac{x}{\sqrt{x^2 + y^2}} & \frac{y}{\sqrt{x^2 + y^2}} \\ \frac{- y}{x^2 + y^2} & \frac{x}{x^2 + y^2} \end{bmatrix} = \begin{bmatrix} \cos \theta & \sin \theta \\ \frac{- \sin \theta}{r} & \frac{\cos \theta}{r} \end{bmatrix}</math>

with determinant 1/r.

with determinant 1/r.

::<math>f_1,f_2,...,f_n \to h_1,h_2,...,h_n</math>

::<math>f_1,f_2,...,f_n \to h_1,h_2,...,h_n</math>

is the result of doing both, then

is the result of doing both, then

::<math>\frac{\partial(h_1,h_2,...,h_n)}{\partial(f_1,f_2,...,f_n)} = \frac{\partial(h_1,h_2,...,h_n)}{\partial(g_1,g_2,...,g_n)}\ \frac{\partial(g_1,g_2,...,g_n)}{\partial(f_1,f_2,...,f_n)}</math>

::<math>\frac{\partial(h_1,h_2,...,h_n)}{\partial(f_1,f_2,...,f_n)} = \frac{\partial(h_1,h_2,...,h_n)}{\partial(g_1,g_2,...,g_n)}\ \ \frac{\partial(g_1,g_2,...,g_n)}{\partial(f_1,f_2,...,f_n)}</math>

Hint:  This is really just the chain rule for partial derivatives.  The adding up of terms in the chain rule is the same as the adding up of products in a matrix multiplication.

Hint:  This is really just the chain rule for partial derivatives.  The adding up of terms in the chain rule is the same as the adding up of products in a matrix multiplication.