Engineering Math - Vector Calculus and Partial Differential Equations

Learn vector calculus fundamentals and their applications to partial differential equations through this comprehensive 9-hour 20-minute course. Master essential vector calculus operations including divergence, gradient, and curl, then apply these concepts to derive and solve fundamental partial differential equations in physics. Explore the mathematical foundations behind conservation laws, starting with basic vector field operations and progressing to advanced topics like Gauss's divergence theorem, Stokes' theorem, and the Helmholtz decomposition. Derive key partial differential equations including the continuity equation for mass conservation, Laplace's equation for potential flow, the heat equation for thermal energy conservation, and the wave equation for oscillatory phenomena. Practice solving these equations using multiple techniques such as separation of variables, the method of characteristics, and Laplace transforms. Examine real-world applications including guitar string physics, slack line dynamics, and heat distribution problems. Gain insight into the mathematical language used to describe physical phenomena and conservation principles across various fields of engineering and physics.

Syllabus

Vector Calculus and Partial Differential Equations: Big Picture Overview
Div, Grad, and Curl: Vector Calculus Building Blocks for PDEs [Divergence, Gradient, and Curl]
The Gradient Operator in Vector Calculus: Directions of Fastest Change & the Directional Derivative
The Divergence of a Vector Field: Sources and Sinks
The Curl of a Vector Field: Measuring Rotation
Gauss's Divergence Theorem
The Continuity Equation: A PDE for Mass Conservation, from Gauss's Divergence Theorem
Stokes' Theorem and Green's Theorem
Are all vector fields the gradient of a potential? ... and the Helmholtz Decomposition
Laplace's Equation and Potential Flow
Potential Flow Part 2: Details and Examples
Partial Differential Equations Overview
Laplace's Equation and Poisson's Equation
Deriving the Heat Equation: A Parabolic Partial Differential Equation for Heat Energy Conservation
The Heat Equation and the Steady State Heat Distribution via Laplace's Equation
Deriving the Heat Equation in 2D & 3D (& in N Dimensions!) with Control Volumes and Vector Calculus
PDE 101: Separation of Variables! ...or how I learned to stop worrying and solve Laplace's equation
Deriving the Wave Equation
Solving the Wave Equation with Separation of Variables... and Guitar String Physics
The Method of Characteristics and the Wave Equation
The Wave Equation and Slack Line Physics
Solving PDEs with the Laplace Transform: The Heat Equation
Solving PDEs with the Laplace Transform: The Wave Equation