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.

[prabhsimransinghbadwal]

Amazing explanation of each and every concept. I love the dedication you show towards teaching.

[ambrishabhijatya]

Professor, might I suggest putting 'Jacobian' in the title. It is usually introduced in the context of multiple integrals without any motivation. I had to browse through umpteen number of videos with such introductions to reach this one.

Diana, at 35:50 you asked what is f evaluated at (1.1, 1.1, 1.1) and the answer is to use the linearization of the function. One thing you said is that it is gonna be easier (which indeed is true). Please correct me if I am wrong: the two values f(1.1, 1.1, 1.1) and L[1.1, 1.1, 1.1] are not the same, but the second one is an approximation of the first. I am thinking of Euler's method, as an analogy (if you will).
By the way: I love the enthusiasm and energy you display at every lecture. You are fantastic as a lecturer.

[tech_science_tutos]

excellent lectures, you must make another lectures in another topics.

[Hindi_poetry]

is there any difference between continuity and

[narwhale]

Hello. Do you think you could drop a link to the materials you give to your students? It'd be very helpful

[jasonarmondo]

great video but grainy for some reason ;p