scalar field f in the space Rn (whose independent coordinates are the components of x) is the transpose of the derivative of a scalar by a vector. By example, in physics, the electric field is the negative vector gradient of the electric potential. The directional derivative of a scalar function f(x) of the space vector x in the direction of the unit vector u (represented in this case as a column vector) is defined using the gradient as follows. Using the notation just defined for the derivative of a scalar with respect to a vector we can re-write the directional derivative as This type of notation will be nice when proving product rules and chain rules that come out looking similar to what we are familiar with for the scalar derivative. Vector-by-vector Each of the previous two cases can be considered as an application of the derivative of a vector with respect to a vector, using a vector of size one appropriately. Similarly we will find that the derivatives involving matrices will reduce to derivatives involving vectors in a corresponding way. The derivative of a vector function (a vector whose components are functions) , with respect to an input vector, , is written (in numerator layout notation) as In vector calculus, the derivative of a vector function y with respect to a vector x whose components represent a space is known as the pushforward (or differential), or the Jacobian matrix. The pushforward along a vector function f with respect to vector v in Rn is given by Derivatives with matrices There are two types of derivatives with matrices that can be organized into a matrix of the same size. These are the derivative of a matrix by a scalar and the derivative of a scalar by a matrix. These can be useful in minimization problems found in many areas of applied mathematics and have adopted the names tangent matrix and gradient matrix respectively after their analogs for vectors. Note: The discussion in this section assumes the numerator layout convention for pedagogical purposes. Some authors use different conventions. The section on layout conventions discusses this issue in greater detail. The identities given further down are presented in forms that can be used in conjunction with all common layout conventions. Matrix-by-scalar The derivative of a matrix function Y by a scalar x is known as the tangent matrix and is given (in