gradient

Gradient vector of scalar function

Syntax

gradient(f,v)

Description

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 = 
$$\left(\begin{array}{c}-2\hspace{0.17em}\cos \left(x\right)\hspace{0.17em}\left(\sin \left(x\right)+\sin \left(y\right)\right)\\ -2\hspace{0.17em}\cos \left(y\right)\hspace{0.17em}\left(\sin \left(x\right)+\sin \left(y\right)\right)\end{array}\right)$$

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

f — Scalar function

symbolic expression | symbolic function

Scalar function, specified as symbolic expression or symbolic function.

v — Vector with respect to which you find gradient vector

symbolic vector

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.

More About

Gradient Vector

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

$$\nabla f=\left(\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2},\dots ,\frac{\partial f}{\partial x_n}\right)$$