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→Partial Derivatives: more
Here are some alternative notations for partial derivatives:
Here are some alternative notations for partial derivatives:
:<math>\frac{\partial}{\partial x} f(x,y,z) = \partial_x f(x,y,z) = f_x (x,y,z)</math>
:<math>\frac{\partial}{\partial x} f(x,y,z) = \partial_x f(x,y,z) = f_x (x,y,z)</math>
One can take higher-order partial derivatives as well, such as
:<math>\frac{\partial^2 f}{\partial x^2}</math>
or
:<math>\frac{\partial^2 f}{\partial x \partial y}</math>
which means
:<math>\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right)</math>
A question that arises is: do mixed partial derivatives commute? That is, do we have:
:<math>\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}</math>
The answer is yes, ''if the derivatives are continuous''. In problems that arise in practice, the derivatives are always continuous, and switching derivative order is a staple of work in multivariate calculus.
=Problems=
=Problems=