Gradient of a scalar function

The gradient of a scalar function (or field) is a vector-valued function directed toward the direction of fastest increase of the function and with a magnitude equal to the fastest increase in that direction. It is denoted with the ∇ symbol (called nabla, for a Phoenician harp in greek). The gradient is therefore a directional derivative.

A scalar function associates a number (a scalar value) to every point of the space.
A vector-valued function (or vector function) associates a vector to every point of the space.

Two scalar fields are represented in the upper figure (the left one has a circular symmetry). As you can see, the field value increases (inward for the left field, and from the right to the left for the right field). The vectors (vector-valued function) represent the gradient and are directed toward the direction of fastest increase of the scalar function.

An example of gradient is for instance the temperature change inside a room. The temperature is a scalar quantity, so we can mathematically represent it as a function f(x,y,z). For any point (x,y,z) of the room, the f function returns a number (the temperature in this point).

To make it simple, we will consider the temperature to be invariant in time. If we calculate the gradient of f at a point (x,y,z), the resulting vector-valued function will return the direction of the fastest increase of temperature at this point. The gradient magnitude will correspond to the rate of the temperature rise in this direction.

The way the gradient is calculated depends on the coordinates system used.

Gradient in Cartesian coordinates:

Where the ∂ symbol denotes the partial derivative of function f with respect to the corresponding variable. When a scalar function depends on multiple variables, its partial derivative with respect to one variable is calculated with the others variables held constant.

i, j and k are the unit vectors for directions x, y and z respectively.

Example of partial derivatives of a scalar function:

And its gradient:

Gradient in spherical coordinates:

The spherical coordinates are represented in the following figure:

And the gradient of function f expressed in spherical coordinates is given by:

Where u_r, u_θ and u_φ are the unit vectors for directions r, θ y φ.

When a function f depends only on the radial coordinate (for instance the electrostatic potential energy), its gradient is given by:

Where the total derivative symbol d is used instead of the partial derivative ∂ because here f depends only on a single variable (r).

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