Partial derivative

A partial derivative of a function of many variables is the derivative of that function with respect to one of its arguments.

For example, if $f(x,y,z) = xy + z(x + y),$ then $f$ has three partial derivatives at the point $(3,4,12)$ :

$\frac{\partial f}{\partial x} = (4x + 12(x + 4))' \rvert_{x = 3} = 16,$
$\frac{\partial f}{\partial y} = (3y + 12(3 + y))' \rvert_{y = 4} = 15,$
$\frac{\partial f}{\partial z} = (3*4 + z(3 + 4))' \rvert_{z = 12} = 7.$

Del operator

The del operator, or nabla symbol, written $\nabla$ , represents the vector $\left( \frac{\partial}{\partial x_1}, \frac{\partial}{\partial x_2}, \dots , \frac{\partial}{\partial x_n} \right)$ where the value $n$ is the arity (number of arguments) of the function in question.

Gradient

The product of $\nabla$ and a function $f(x_1, x_2, \dots, x_n)$ is $\nabla f = \left( \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \dots , \frac{\partial f}{\partial x_n} \right),$ a vector storing, in order, all of the partial derivatives of $f$ .

The gradient applies when $f$ is a scalar-valued function of many variables. For example, the gradient of temperature in a closed room is $\left( \frac{\partial T}{\partial x}, \frac{\partial T}{\partial y}, \frac{\partial T}{\partial z} \right)$ , where $x$ , $y$ , and $z$ are the Cartesian coordinates in the three spatial dimensions: length, width, and height, respectively. $\frac{\partial T}{\partial x}$ and $\frac{\partial T}{\partial y}$ are likely to be close to zero at most points, but $\frac{\partial T}{\partial z}$ probably has a small positive value, since the air nearer the ceiling (greater $z$ ) is warmer than the air nearer the floor (lesser $z$ ). Therefore, the typical direction of the gradient vector $\nabla T$ is close to upwards.

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Partial derivative

Contents

Del operator

Gradient

Divergence

Curl

Applications

See also