Advanced Differential Calculus

Video Lectures

Displaying all 7 video lectures.
Lecture 1
Partial Derivatives Review
Play Video
Partial Derivatives Review


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.

Lecture 2
Using Four or More Variables
Play Video
Using Four or More Variables


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.

Lecture 3
Evaluation of Partial Derivatives
Play Video
Evaluation of Partial Derivatives


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.

Lecture 4
The Total Differential
Play Video
The Total Differential


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

Lecture 5
The idea of delta(z) and dz
Play Video
The idea of delta(z) and dz


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

Lecture 6
Theorem of the Total Differential
Play Video
Theorem of the Total Differential


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

Lecture 7
Fundamental Lemma
Play Video
Fundamental Lemma


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.