Gradient

### 13.7.1 Gradient

The derive command finds partial derivatives of a multivariable expression. diff and grad can be used synonymously for derive here.

- derive takes two arguments:
  - expr, an expression involving n real variables.
  - [x1,…,xn], a vector of the variable names.
- derive(expr,[x1,…,xn]) returns the gradient of expr; namely, the vector of partial derivatives of expr with respect to x1, …, xn.

  For instance, in dimension n=3, with variables [x,y,z]:

  | ▸
  grad
  | (F)= | ⎡ ⎢ ⎢ ⎣ |
  ∂ F
  ∂ x
  | , |
  ∂ F
  ∂ y
  | , |
  ∂ F
  ∂ z
  | ⎤ ⎥ ⎥ ⎦ | .

#### Example

Find the gradient of F(x,y,z)=2x2y−xz3.

derive(2*x^2*y-x*z^3,[x,y,z])

or:

diff(2*x^2*y-x*z^3,[x,y,z])

or:

grad(2*x^2*y-x*z^3,[x,y,z])

| ⎡ ⎣ | 2· 2 x y−z3,2 x2,−3 x z2 | ⎤ ⎦

normal(ans())

| ⎡ ⎣ | 4 x y−z3,2 x2,−3 x z2 | ⎤ ⎦

To find the critical points of F(x,y,z)=2x2y−xz3:

solve(derive(2*x^2*y-x*z^3,[x,y,z]),[x,y,z])

| ⎡ ⎣ | ⎡ ⎣ | 0,y,0 | ⎤ ⎦ | ⎤ ⎦