Higher Order Partial Derivatives

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Colloquium

In connection with the math research weekend, so related to “Dynkin diagrams, quaternions, and the highest root game.”

By Prof. Allen Knutson, professor of math, Cornell University.

Friday (October 14), 4:00 - 5:00 PM.

Newton 204.

2nd (and Higher) Partial Derivatives

Based on “Higher-Order Partial Derivatives” in section 3.3 of the textbook.

Key Ideas

Higher-order partial derivatives are partial derivatives of partial derivatives (analogous to higher-order derivatives of single-variable functions, except that different derivatives in the sequence can be with respect to different variables)

Clairaut’s Theorem says mixed derivatives involving the same variables are equal, regardless of the order of the variables, as long as those derivatives are continuous.

Examples

Find some second and third partial derivatives z = sinx cosy

It’s often helpful to start by finding the first partial derivatives:

Then differentiate the first derivatives with respect to the relevant variables to find second derivatives:

And you can use second derivatives to find third derivatives, and so forth:

Suppose f(x,y,z) = √(x² + y² + z²). What are some of its higher order derivatives?

We only had time to find one of the second derivatives (and then to use Clairaut’s Theorem, which applies to more than 2 variables just as it does to 2, to find a derivative with respect to the same variables in a different order).

Next

Tangents to surfaces.

Please read “Tangent Planes,” “Linear Approximations,” and “Differentiability” in section 3.4 of the textbook.

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