A pdf version of this post is *here*.

The time velocity of an *n*-dimensional vector variable or vector-valued function Δ**x** per unit of an independent scalar variable *t* equals

Similarly, the space lenticity with Δ**t** and Δ*x*, respectively:

The rate of a scalar variable Δ*x* per unit of an independent *n*-dimensional vector variable or vector-valued function Δ**t** equals

The same except with Δ*t* and Δ**x**, respectively:

The rate of an *n*-dimensional vector variable or vector-valued function Δ**x** per unit of an independent *n*-dimensional vector variable or vector-valued function Δ**t** equals

The same except with Δ**t** and Δ**x**, respectively:

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

The same except with **t** and *x*, respectively:

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

The same except with *t* and **x**, respectively:

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

The same except with **t** and **x**, respectively:

The matrices are square Jacobian matrices, whose (*i, j*)^{th} entry is

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.