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Brilliant

Vector Calculus

via Brilliant

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Overview

Change is deeply rooted in the natural world. Fluids, electromagnetic fields, the orbits of planets, the motion of molecules; all are described by vectors and all have characteristics depending on where we look and when.

In this course, you'll learn how to quantify such change with calculus on vector fields. Go beyond the math to explore the underlying ideas scientists and engineers use every day.

Syllabus

  • Introduction: Vector fields, surface integrals, div and curl.
    • Vector Calculus in a Nutshell: Explore the possibilities that come from combining calculus and vectors.
  • Vector-valued Functions: Force fields, motion through space, and much, much more...
    • Calculus of Motion: Look at the world in motion through the lens of vector calculus.
    • Space Curves: Fly along curves through three dimensions.
    • Integrals and Arc Length: Exactly how long is a space curve?
    • Frenet Formulae: Measure the shape of space curves with vector calculus.
    • Parametric Surfaces: Expand your gallery of shapes to include a few exotic cases.
    • Vector Fields: Combine vectors and functions into a powerful tool for applications.
    • Jack and the Beanstalk: Learn about Newton's root-finding method and help defeat a vile giant.
    • Electrostatic Bootcamp: Take a lightning tour of the physics that made vector calculus famous.
  • Surface Integrals and Divergence: Part I of the essential vector calculus toolbox.
    • Introducing Surface Integrals: Mix vectors with integrals to uncover an essential tool for applications.
    • Flux (Part I): Experiment with charges moving in electric field and discover the concept of flux.
    • Flux (Part II): Use flux to uncover surface integrals and see how they're used to solve important problems.
    • Surface Integrals: Master integrals of functions on parametrized surfaces.
    • Divergence (Part I): Construct a crucial vector derivative through a fundamental law of nature.
    • Divergence (Part II): What is divergence?
    • The Divergence Theorem: Prove a beautiful integral identity that's essential for real-world applications.
    • The Divergence Theorem and Fluids: Another view of one of the most important theorems in vector calculus.
    • Flows & Divergence: Explore more about the divergence through visuals and geometry.
  • Line Integrals & Curl: Part II of the essential vector calculus toolbox.
    • Work (Part I): Explore an important physics application of vector calculus.
    • Work (Part II): Unveil a new kind of integral by delving into a familiar physics concept.
    • Line Integrals: Learn how to integrate along space curves and why it's so useful.
    • Path Independence: Journey to where calculus and topology meet to discover a crucial property of vector fields.
    • Curl: Construct a derivative that measures the swirl of a vector field.
    • Stokes' Theorem: Uncover the deep connection between curl and line integrals.
    • Swirls and Curls: Dive deeper into curl with visuals and geometry.
    • Differential Forms (Optional): Unify vector calculus into a single master formula.
  • Advanced Applications: Solve important real-world problems with vector calculus.
    • The Laplacian: Apply a new derivative to problems in electrostatics and fluid dynamics.
    • Gaussian Integrals (Part I): Detour into the world of multivariable calculus to compute an integral crucial for applications.
    • Gaussian Integrals (Part II): Use linear algebra and a change of variables to find the most general Gaussian integral.
    • Fourier Transform: Apply Gaussian integrals to understand the Fourier transform, a powerful way to solve pdes.
    • The Diffusion Equation: Use the Fourier transform to solve a pde and calculate the speed of scent.
    • The Wave Equation: Make waves with Fourier series.
    • Maxwell's Equations (Part I): Combine divergence, curl, line and surface integrals to uncover Maxwell's electromagnetic equations.
    • Maxwell's Equations (Part II): Use vector calculus to solve electromagnetic problems and unify Maxwell's equations with forms.
    • More Electrostatics!: Squeeze new pde methods from old vector calculus theorems and solve hard electrostatics problems.

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