Vector Calculus

Course Overview

  • Line integrals
  • Surface integrals
  • Conservative vector fields
  • Divergence and Curl

Course Goals

  • For students to develop a geometric understanding of vectors (without components), including the dot and cross products.
  • For students to develop the ability to express vectors in standard coordinate systems and bases.
  • For students to develop a geometric understanding of the gradient, including its relationship to level sets.
  • For students to develop a geometric understanding of conservative vector fields, including their relationship to the gradient and to level sets.
  • For students to develop the ability to evaluate line and surface integrals;
  • For students to develop a geometric understanding of the curl and divergence, including their relationship to circulation and flux.
  • For students to develop the ability to evaluate the curl and divergence of a vector field in standard coordinate systems.
  • For students to develop a geometric understanding of the Divergence Theorem and Stokes' Theorem.
  • For students to appreciate the unifying thread brought to these topics by the use of a “use what you know” strategy starting from the vector differential.

Sample Syllabi

Course Contents

Introduction

Vectors
Differentials and Derivatives
Surfaces
Gradient and Directional Derivatives
Polar and Curvilinear Coordinates

Line Integrals

Line Integrals
Conservative Vector Fields

Surface Integrals

Surfaces
Flux

Vector Derivative Operators

Divergence and Curl
Theorems
Applications

Activities Included


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