Difference between revisions of "Partial derivative"
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Revision as of 10:29, 3 May 2022
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 
then 
has three partial derivatives at the point 
:
Del operator
The del operator, or nabla symbol, written 
, represents the vector ![\[\left( \frac{\partial}{\partial x_1}, \frac{\partial}{\partial x_2}, \dots , \frac{\partial}{\partial x_n} \right)\]](http://latex.artofproblemsolving.com/8/6/e/86e44828adbbf7ec1e319339fa7a1b9a3fafbb2a.png)
where the value 
is the arity (number of arguments) of the function in question.
Gradient
The product of and a function 
is ![\[\nabla f = \left( \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \dots , \frac{\partial f}{\partial x_n} \right),\]](http://latex.artofproblemsolving.com/f/d/7/fd744045e9b83ca619907644c18532c9f41adcf5.png)
a vector storing, in order, all of the partial derivatives of .
The gradient applies when is a scalar-valued function of many variables. For example, the gradient of temperature in a closed room is 
, where 
, 
, and 
are the Cartesian coordinates in the three spatial dimensions: length, width, and height, respectively. 
and 
are likely to be close to zero at most points, but 
probably has a small positive value, since the air nearer the ceiling (greater ) is warmer than the air nearer the floor (lesser ). Therefore, the typical direction of the gradient vector 
is close to upwards.
