Gradient

The gradient for a function of several variables is a vector-valued function whose components are partial derivatives of those variables. The gradient can be thought of as the direction of the function's greatest rate of increase.

Formally, given a multivariate function f with n variables and partial derivatives, the gradient of f, denoted ∇f, is the vector valued function,

where the symbol ∇, named nabla, is the partial derivative operator. For example, to find the gradient, ∇f(1, 2, 3) for f(x, y, z) = 4x²yz² + 2xy² - xyz, take the partial derivatives of x, y, and z:

Substituting 1, 2, and 3 in for x, y, and z then yields:

Properties of the gradient

Let y = f(x, y) be a function for which the partial derivatives f_x and f_y exist.

• If the gradient for f is zero for any point in the xy plane, then the directional derivative of the point for all unit vectors is also zero. That is, if ∇f(x, y) = 0, then D_u(x, y) = 0 for any u.


• The directional derivative for any point in the xy plane has its maximum increase when it is in the direction of its gradient. Its maximum value is the magnitude of its gradient. That is, if ∇f(x, y) ≠ 0, then the maximum of D_u(x, y) is ||∇f(x, y)||.


• The minimum value for the directional derivative at any point in the xy plane is -||∇f(x, y)|| in the direction of -∇f(x, y).

Geometric interpretation of the gradient for a function of two variables

Consider the following graph with gradient vectors denoted in red. The graph of z = f(x, y) is a paraboloid opening upward along the z-axis whose vertex is at the origin.

The gradient vectors ∇f(x₁, y₁) and ∇f(x₂, y₂) drawn in the xy-plane have their initial points placed at (x₁, y₁) and (x₂, y₂) respectively. Note that the gradient vectors are calculated in component form but are translated to their respective input points on the xy plane to better show the direction associated with those points. If we could fold the xy plane so each input (x, y) maps to their corresponding output value (x, y, z) on the paraboloid, then each corresponding gradient vector can also be mapped onto the paraboloid such that its initial point is at (x, y, z). The gradient vectors mapped to (x₁, y₁, z₁) and (x₂, y₂, z₂) show the direction of fastest increase.

Gradient vector field

Finding the gradient for each point in the xy plane in which a function f(x, y) is defined creates a set of gradient vectors called a gradient vector field. The gradient vector field gives a two-dimensional view of the direction of greatest increase for a three-dimensional figure. A gradient vector field for the paraboloid graphed above is shown below:

The equation of the paraboloid above is f(x, y) = 0.3x² + 0.3y². The equation of the vector field is:

Notice that the vectors lengthen as they move away from the origin, confirming that the surface of the paraboloid gets steeper the further from the origin it gets.

Using gradients to find directional derivatives

The directional derivative of a function z = f(x, y) in the direction of the unit vector , where the partial derivatives f_x and f_y exist, is:

If the direction is given as an angle θ in standard position, then , and