In this course, Multivariable Calculus is explained by focusing on understanding the key concepts rather than learning the formulas and/or exercises by rote. The process of reasoning by using mathematics is the primary objective of the course, and not simply being able to do computations. Besides, interesting proofs will be given, such as the Gauss and Stokes theorems proofs.
The prior knowledge requirement is Single variable Calculus (even without a great mastery of it).
I will list some of the most important concepts that we will see here in the following.
partial differentiation. The partial derivative generalizes the notion of the derivative to higher dimensions. A partial derivative of a multivariable function is a derivative with respect to one variable with all other variables held constant. Partial derivatives may be combined in interesting ways to create more complicated expressions of the derivative. For example, in vector calculus (which we will see), the "del" operator is used to define the concepts of gradient, divergence, and curl in terms of partial derivatives. A matrix of partial derivatives, the Jacobian matrix, may be used to represent the derivative of a function between two spaces of arbitrary dimension. Differential equations containing partial derivatives are called partial differential equations or PDEs. These equations are generally more difficult to solve than ordinary differential equations, which contain derivatives with respect to only one variable (PDEs are not discussed in this course).
Multiple integration. The multiple integral extends the concept of the integral to functions of any number of variables. Double and triple integrals may be used to calculate areas and volumes of regions in the plane and in space.
The surface integral and the line integral are used to integrate over curved manifolds such as surfaces and curves. We will see these concepts.
I am available for questions, which I could answer by (possibly) uploading new content to the course, namely videos containing the solution.
Concept of double integral
FREE PREVIEWDouble integral in Cartesian coordinates
FREE PREVIEWChange of variables in double/triple integrals
FREE PREVIEWDouble integral in Polar coordinates
Triple integral in spherical coordinates
Surface integrals part 1
Surface integrals part 2
Surface integrals part 3
Divergence theorem and its proof
Line integrals part 1
Line integrals part 2
Stokes theorem and its proof
Application of Stokes theorem
Another application of Stokes theorem
Corollary of the Divergence Theorem
divergence theorem to calculate the surface of a sphere
FREE PREVIEWdivergence theorem to calculate the surface of a cube
surface area of a tetrahedron by surface integral and then divergence theorem
Double integral: changing the limits of integration ex 1
FREE PREVIEWDouble integral: changing the limits of integration ex 2
double integral ex 1
Triple integral ex 1: Volume delimited by paraboloid and sphere
Triple integral 2
Double integral to calculate a famous series (Basel problem)
line and surface integrals ex 1
line and surface integrals ex 2
Surface integral ex 1 (solved without resorting to the divergence theorem)
surface integral ex 1 solved via divergence theorem
surface integral ex 2: calculation of the area of a portion of a plane