Vectors and functions

Vectors and functions is here in pdf form.

The quotient of an n-dimensional vector variable or vector-valued function x with a scalar variable t equals

$\frac{\mathbf{x}}{t}=\left (\frac{x_1}{t}, \frac{x_2}{t},...,\frac{x_n}{t} \right ).$

The quotient of a scalar variable x with an n-dimensional vector variable or vector-valued function t equals

$\frac{t}{\mathbf{x}}=\left (\frac{t}{x_1}, \frac{t}{x_2},...,\frac{t}{x_n} \right ).$

Note: transpose these vectors for numerator notation.

The quotient of an n-dimensional vector variable or vector-valued function x with an independent n-dimensional vector variable or vector-valued function t equals

$\frac{\mathbf{x}}{\mathbf{t}}=\begin{pmatrix} \frac{x_1}{t_1} & ... & \frac{x_1}{t_n} \\ ... & ... & ... \\ \frac{x_n}{t_1} & ... & \frac{x_n}{t_n} \end{pmatrix}$

The derivative of an n-dimensional vector function x with respect to an independent scalar variable t equals

$\frac{d\mathbf{x}}{dt}=\left (\frac{\partial x_1}{\partial t}, \frac{\partial x_2}{\partial t},...,\frac{\partial x_n}{\partial t} \right ).$

The directional derivative of a scalar function x with respect to an independent n-dimensional vector variable t equals

$\frac{dt}{d\mathbf{x}}=\left (\frac{\partial t}{\partial x_1}, \frac{\partial t}{\partial x_2},...,\frac{\partial t}{\partial x_n} \right ).$

Note: transpose these vectors for denominator notation.

The directional derivative of an n-dimensional vector-valued function x with respect to an n-dimensional vector variable t equals

$\frac{\partial \mathbf{x(\mathbf{t})}}{\mathbf{t}}=\begin{pmatrix} \frac{\partial x_1}{t_1} & ... & \frac{\partial x_1}{t_n} \\ ... & ... & ... \\ \frac{\partial x_n}{t_1} & ... & \frac{\partial x_n}{t_n} \end{pmatrix}$

The matrix is a square Jacobian matrix, whose (i, j)^th entry is

$\mathbf{J}_{ij} =\frac{\partial x_i}{\partial t_j} \: or \: \frac{\partial t_i}{\partial x_j}.$

The inverse function theorem states that the matrix inverse of the Jacobian matrix of an invertible function is the Jacobian matrix of the inverse function.