Multivariable/Multi-Variable/Multivariate Calculus/Analysis

Multi-Variate Calculus - Derivative - Preliminary

• Derivative of Scalar-Valued Function (Partial Derivative - Total Derivative - Gradient - Directional Directive - Second Order Partial Derivative)
• Derivative of Vector-Valued Function (Partial Derivative)

Multi-Variate Calculus - Derivative - Building Blocks

Video Lectures: Div, Grad, and Curl: Vector Calculus Building Blocks for PDEs [Divergence, Gradient, and Curl]

Multi-Variate Calculus - Derivative - Compound Blocks

Multi-Variate Calculus - Derivative - Applications

TODO https://www.khanacademy.org/math/multivariable-calculus/applications-of-multivariable-derivatives

Multi-Variate Calculus - Integral

TODO https://www.khanacademy.org/math/multivariable-calculus/integrating-multivariable-functions

• Curvilinear Integral
• Surface Integral
• Volume Integral

Multi-Variate Calculus - The Fundamental Theorem of Calculus, Green’s Theorem, Divergence Theorem, Stoke’s Theorem

1. Third Fundamental Theorem of Calculus (Part I, II, and III)
2. Green’s Theorem
3. 2D Divergence Theorem & 3D Divergence Theorem
4. Stokes’ Theorem

They are the same theorem. Once you have the notion of differential forms, define the operator 𝑑 which takes a form into a form of higher degree, and create the notion of integration of a form on an oriented manifold, you obtain Stokes’ theorem which says:

• ${\int }_{X}d𝜔={\int }_{\partial X}𝜔$

If 𝜔 is a 0-form, this is the fundamental theorem of calculus
If 𝜔 is a 1-form, this is Green’s theorem

Green’s theorem on the plane can be expressed in two forms, the flux-divergence form and the circulation-curl form.

The divergence theorem is the extension of the flux-divergence form to a closed, orientable 3D surface, and Stoke’s theorem is the extension of the circulation-curl form.

Resources

• https://www.khanacademy.org/math/multivariable-calculus
• Steve Brunton’s Engineering Math: Vector Calculus and Partial Differential Equations