Multivariable calculus, class #8: Linearization, the Jacobian, and higher-order partial derivatives

Mathematician spotlight: Ryan Hynd We review the idea of linear approximation (tangent plane) for functions from R^2 to R, now writing it in vector form, and define the gradient. We then extend the notion of best linear approximation to functions from R^m to R^n, and define the Jacobian matrix. We do an example. Finally, we take all four second partial derivatives of a function of two variables, and state and discuss Clairaut's Theorem.