Advanced Differential Calculus

In this video lesson, Calculus Instructor Donny Lee teaches about Partial Derivatives. Gaussianmath ventures back into pure mathematics with it's first analysis course focusing on the advanced concepts in differential calculus.



In this module, we work from the ground up using basic definitions and first principles to derive famous theorems in calculus. We question to the core why certain things are the way they are.

In this video lesson, Calculus Instructor Donny Lee teaches about quick clarification on variables. When only three variables x, y, z are involved, the notation for partial derivatives is self-explanatory: x and y are independent, and we differentiate holding one of them fixed.



With four or more variables, we need to be careful in specifying which are the independent variables. u = h(x, y, v) and u = f(x, y); v = g(x, y) mean two different things.

In this video lesson, Calculus Instructor Donny Lee teaches about the Evaluation of Partial Derivatives. Evaluation of partial derivatives is carried out as in ordinary calculus, for one is always differentiating a function of one variable, the other being treated as a constant.

In this video lesson, Calculus Instructor Donny Lee teaches about the Total Differential. Previously for partial derivatives, â

In this video lesson, Calculus Instructor Donny Lee teaches about the idea of delta(z) and dz

In this video lesson, Calculus Instructor Donny Lee teaches about the Theorem of the Total Differential. In fact, there is a quick way in calculating the total differential of a function IF it exist. Here, we see how we get to the result: a = ∂z / ∂x, b = ∂z / ∂y

In this video lesson, Calculus Instructor Donny Lee teaches about the Fundamental Lemma. In discussing about the concept of the total differential, there should be certain hypotheses which, when satisfied by a function, tell us whether a function has a total differential or is differentiable at a point (x, y).



This 'fundamental lemma' is just that and it is based on the continuity of partial derivatives.