The Gradient of a Function

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In one of my previous posts we talked about partial derivatives for multivariable functions. For each independent variable of a function we can take the partial derivative of the function with respect to that variable. The gradient of a function is another way to take a derivative of a multivariable function which takes in to account all of the partial derivatives of the function.

The symbol for the gradient of a function is ∇ and we will denote the gradient of the function f by ∇f. The gradient of f(x, y, z) is calculated as follows:

∇f = (f_x, f_y, f_z)

To find the gradient of a function we take its partial derivative with respect to each variable and make it the coordinate of a vector in the proper order. The definition does not depend on f being a function of three variables as above but it can be calculated for a function of any number of variables. For example, if we have a function of one variable then the gradient just becomes the normal derivative of the function.

We will now do a couple of examples of how to calculate the gradient of a function. Suppose that f(x, y) = x² + y² then we have that f_x = 2x and f_y = 2y so the gradient of f is

∇f = (2x, 2y)

Note that this gives us the gradient at any point for which it is defined. We can find the value of the gradient at a particular point by plugging the values in for x and y. For example, the value of the gradient at the point (1, 1) would just be (2, 2) in this case.

In the next example let us consider the function of three variables f(x, y, z) = x - zcos(y). The partial derivatives of this function are f_x = 1, f_y = zsin(y) and f_z = -cos(y) so it follows that the gradient of f is

∇f = (1, zsin(y), -cos(y))

The value of the gradient at the point (0, π, 1) is then (1, 0, 1).

The gradient can be thought of as an operator that acts on the function f. By abuse of notation we can define

∇ = (∂ / ∂x, ∂ / ∂y, ∂ / ∂z)

to be the vector of partial differentiation operators. Then we can think of the gradient of f as product of this operator and f componentwise as follows

∇f = (∂ / ∂x, ∂ / ∂y, ∂ / ∂z)f = (∂f / ∂x, ∂f / ∂y, ∂f / ∂z)

The gradient of a function acts like a derivative and is used for calculating things such as directional derivatives which will be discussed in another post. Furthermore, the gradient allows us to find the directions in which the maximum and minimum rates of change of the function occur.

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References:

https://en.wikipedia.org/wiki/Gradient
http://mathworld.wolfram.com/Gradient.html

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All content in this post was created by myself.