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Vector này được gọi là '''[[gradient]]''' của ''f'' tại ''a''. Nếu ''f'' khả vi tại mọi điểm trong một miền nào đó, thì gradient là hàm số có trị là vectơ ∇''f'' đưa điểm ''a'' đến vectơ ∇''f''(''a''). Do đó gradient là một
[[trường vectơ]].
==Examples==
[[Image:Cone 3d.png|thumb|The volume of a cone depends on height and radius]]
The [[volume]] ''V'' of a [[cone (geometry)|cone]] depends on the cone's [[height]] ''h'' and its [[radius]] ''r'' according to the formula
:<math>V(r, h) = \frac{\pi r^2 h}{3}.</math>
The partial derivative of ''V'' with respect to ''r'' is
:<math>\frac{ \partial V}{\partial r} = \frac{ 2 \pi r h}{3},</math>
which represents the rate with which a cone's volume changes if its radius is varied and its height is kept constant. The partial derivative with respect to ''h'' is
:<math>\frac{ \partial V}{\partial h} = \frac{\pi r^2}{3},</math>
which represents the rate with which the volume changes if its height is varied and its radius is kept constant.
By contrast, the [[total derivative|''total'' derivative]] of ''V'' with respect to ''r'' and ''h'' are respectively
:<math>\frac{\operatorname dV}{\operatorname dr} = \overbrace{\frac{2 \pi r h}{3}}^\frac{ \partial V}{\partial r} + \overbrace{\frac{\pi r^2}{3}}^\frac{ \partial V}{\partial h}\frac{\operatorname d h}{\operatorname d r}</math>
and
:<math>\frac{\operatorname dV}{\operatorname dh} = \overbrace{\frac{\pi r^2}{3}}^\frac{ \partial V}{\partial h} + \overbrace{\frac{2 \pi r h}{3}}^\frac{ \partial V}{\partial r}\frac{\operatorname d r}{\operatorname d h}</math>
The difference between the total and partial derivative is the elimination of indirect dependencies between variables in partial derivatives.
If (for some arbitrary reason) the cone's proportions have to stay the same, and the height and radius are in a fixed ratio ''k'',
:<math>k = \frac{h}{r} = \frac{\operatorname d h}{\operatorname d r}.</math>
This gives the total derivative with respect to ''r'':
:<math>\frac{\operatorname dV}{\operatorname dr} = \frac{2 \pi r h}{3} + \frac{\pi r^2}{3}k</math>
Which simplifies to:
:<math>\frac{\operatorname dV}{\operatorname dr} = k\pi r^2</math>
Similarly, the total derivative with respect to ''h'' is:
:<math>\frac{\operatorname dV}{\operatorname dh} = \pi r^2</math>
Equations involving an unknown function's partial derivatives are called [[partial differential equation]]s and are common in [[physics]], [[engineering]], and other [[science]]s and applied disciplines.
==Notation==
For the following examples, let ''f'' be a function in ''x'', ''y'' and ''z''.
First-order partial derivatives:
:<math>\frac{ \partial f}{ \partial x} = f_x = \partial_x f.</math>
Second-order partial derivatives:
:<math>\frac{ \partial^2 f}{ \partial x^2} = f_{xx} = \partial_{xx} f.</math>
Second-order [[mixed derivatives]]:
:<math>\frac{\partial^2 f}{\partial y \, \partial x} = \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) = f_{xy} = \partial_{yx} f.</math>
Higher-order partial and mixed derivatives:
:<math>\frac{ \partial^{i+j+k} f}{ \partial x^i\, \partial y^j\, \partial z^k } = f^{(i, j, k)}.</math>
When dealing with functions of multiple variables, some of these variables may be related to each other, and it may be necessary to specify explicitly which variables are being held constant. In fields such as [[statistical mechanics]], the partial derivative of ''f'' with respect to ''x'', holding ''y'' and ''z'' constant, is often expressed as
:<math>\left( \frac{\partial f}{\partial x} \right)_{y,z}.</math>
==Antiderivative analogue==
There is a concept for partial derivatives that is analogous to [[antiderivative]]s for regular derivatives. Given a partial derivative, it allows for the partial recovery of the original function.
Consider the example of <math>\frac{\partial z}{\partial x} = 2x+y</math>. The "partial" integral can be taken with respect to ''x'' (treating ''y'' as constant, in a similar manner to partial derivation):
:<math>z = \int \frac{\partial z}{\partial x} \,dx = x^2 + xy + g(y)</math>
Here, the [[Constant of integration|"constant" of integration]] is no longer a constant, but instead a function of all the variables of the original function except ''x''. The reason for this is that all the other variables are treated as constant when taking the partial derivative, so any function which does not involve <math>x</math> will disappear when taking the partial derivative, and we have to account for this when we take the antiderivative. The most general way to represent this is to have the "constant" represent an unknown function of all the other variables.
Thus the set of functions <math>x^2 + xy + g(y)</math>, where ''g'' is any one-argument function, represents the entire set of functions in variables ''x'',''y'' that could have produced the ''x''-partial derivative 2''x''+''y''.
If all the partial derivatives of a function are known (for example, with the [[gradient]]), then the antiderivatives can be matched via the above process to reconstruct the original function up to a constant.
==See also==
<div style="-moz-column-count:2; column-count:2">
*[[d'Alembertian operator]]
*[[Chain rule]]
*[[Curl (mathematics)]]
*[[Directional derivative]]
*[[Divergence]]
*[[Exterior derivative]]
*[[Gradient]]
*[[Jacobian matrix and determinant]]
*[[Laplacian]]
*[[Symmetry of second derivatives]]
*[[Triple product rule]], also known as the cyclic chain rule.
</div>
==Notes==
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