Partial derivative, Geometry

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Summary

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.

Details

The volume V of a cone depends on the cone's height h and its radius r according to the formula

V(r, h) = \frac3}.

The partial derivative of V with respect to r is

\frac\partial r} = \frac3},

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

\frac\partial h} = \frac3},

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 of V with respect to r and h are respectively

\frac\operatorname dr} = \overbrace\frac\partial r} + \overbrace\frac\partial h}\frac\operatorname d r}

and

\frac\operatorname dh} = \overbrace\frac\partial h} + \overbrace\frac\partial r}\frac\operatorname d h}

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,

k = \fracr} = \frac\operatorname d r}.

This gives the total derivative with respect to r:

\frac\operatorname dr} = \frac3} + \frac3}k

which simplifies to:

\frac\operatorname dr} = k\pi r^2

Similarly, the total derivative with respect to h is:

\frac\operatorname dh} = \pi r^2

The total derivative with respect to both r and h is given by the Jacobian matrix, which here takes the form of the gradient vector \nabla V = \left(\frac\partial r},\frac\partial h}\right) = \left(\frac3}\pi rh, \frac3}\pi r^2\right).

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External Links

  • WikipediaPartial Derivatives

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