# 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.