Documentation

Gradient vector of scalar function

Description

example

gradient(f,v) finds the gradient vector of the scalar function f with respect to vector v in Cartesian coordinates.

If you do not specify v, then gradient(f) finds the gradient vector of the scalar function f with respect to a vector constructed from all symbolic variables found in f. The order of variables in this vector is defined by symvar.

Examples

Find Gradient of Function

The gradient of a function f with respect to the vector v is the vector of the first partial derivatives of f with respect to each element of v.

Find the gradient vector of f(x, y, z) with respect to vector [x, y, z]. The gradient is a vector with these components.

syms x y z
f = 2*y*z*sin(x) + 3*x*sin(z)*cos(y);
gradient(f, [x, y, z])
ans =
3*cos(y)*sin(z) + 2*y*z*cos(x)
2*z*sin(x) - 3*x*sin(y)*sin(z)
2*y*sin(x) + 3*x*cos(y)*cos(z)

Plot Gradient of Function

Find the gradient of a function f(x, y), and plot it as a quiver (velocity) plot.

Find the gradient vector of f(x, y) with respect to vector [x, y]. The gradient is vector g with these components.

syms x y
f = -(sin(x) + sin(y))^2;
g = gradient(f, [x, y])
g =
-2*cos(x)*(sin(x) + sin(y))
-2*cos(y)*(sin(x) + sin(y))

Now plot the vector field defined by these components. MATLAB® provides the quiver plotting function for this task. The function does not accept symbolic arguments. First, replace symbolic variables in expressions for components of g with numeric values. Then use quiver:

[X, Y] = meshgrid(-1:.1:1,-1:.1:1);
G1 = subs(g(1), [x y], {X,Y});
G2 = subs(g(2), [x y], {X,Y});
quiver(X, Y, G1, G2) Input Arguments

collapse all

Scalar function, specified as symbolic expression or symbolic function.

Vector with respect to which you find gradient vector, specified as a symbolic vector. By default, v is a vector constructed from all symbolic variables found in f. The order of variables in this vector is defined by symvar.

If v is a scalar, gradient(f,v) = diff(f,v). If v is an empty symbolic object, such as sym([]), then gradient returns an empty symbolic object.

collapse all

The gradient vector of f(x) with respect to the vector x is the vector of the first partial derivatives of f.

$\nabla f=\left(\frac{\partial f}{\partial {x}_{1}},\frac{\partial f}{\partial {x}_{2}},\dots ,\frac{\partial f}{\partial {x}_{n}}\right)$