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# Resource Guide to Vector Calculus

Vector calculus is a form of mathematics that is focused on the integration of vector fields. Vector calculus uses extensive variations of mathematics from differential geometry to multivariable calculus. Those studying vector calculus must have an understanding of the essential processes and relationships of vector calculus. The following will provide guidance to the operations, theorems, and characteristics of vector calculus.

Mathematical Notation

Vector Notation – Table of common calculus notations.

Integration Techniques
Integration by Parts– Explains this fundamental formula in vector calculus.

Integration of Rational Functions– Examples assessing a method of integrating rational function.

Definite Integrals– Summary and study guide of the properties of the definite integral.

Coordinate Systems in Two and Three Dimensions
Coordinate Systems in Two and Three Dimensions– Introduction and examples to the Cartesian coordinate system.

Vector and Coordinate Systems – Explains the many methods of vectors and coordinate systems.

Vectors
Introduction to Vectors– Provides the basics to vectors.
Dot Product – Lesson plan and practice section on dot products.

Cross Product– Summary and instruction of the vector product.

Equations of Lines and Planes
Line, Planes, and Vectors – Tutorial on the line, planes and vector process.

Vector Equations– Provides detailed direction on finding an equation of a line.

Functions of Two or More Variables
Introduction to Function of Two or More Variables -A concise guide to the introduction to multivariable calculus.
Limits and Continuity– Tutorial for those intimidated by any form of calculus.
Partial Derivatives– Notes on partial derivatives of function of two or more variables.
Gradient and Directional Derivatives– A thorough lecture on gradient and directional derivatives.
Tangent Planes and Differentials– This professor provides lecture notes on four topics of tangent planes and differentials.
Chain Rule for Functions of Two or More Variables– An interactive applet that explores the multivariable chain rule.

Integration

Double Integrals in Rectangular Coordinates – Worksheet using Maple to calculate double integrals.
Double Integrals in Polar Coordinates – Describes the process of converting regular coordinates to polar coordinates.

Triple Integrals– Evaluation and solution of this multiple integral.
Triple Integrals in Cylindrical and Spherical Coordinates– Notes on integration of double and triple integrals in Cartesian coordinates.

Surface Area– Dr. Lou Talman of Metro State lecture on the calculus of surface area.

Vector Functions
Introduction to Vector Functions– The Montana State University math department provides the preamble to groups and vector spaces by defining vector functions with examples of the mathematical function.

Derivative, Unit Tangent Vector, and Arc Length– This lesson explains this function with explicit diagrams and illustrations.
Curvature– Explains how curvature is computed for a particular osculating plane.
Velocity and acceleration – Describes the component form for velocity and acceleration.

Gradients and Directional Derivatives
Gradients and Directional Derivatives– Animated 3-D graphics of directional derivatives and gives details to how gradient vectors coexist within the dot product.

Derivatives– A complete lesson on vectors with section two and three dedicated to gradient and directional derivatives.

Directional Derivatives and the Gradient – Provides examples for finding directional derivatives and defining the gradient function.

Maximization and Minimization of Functions of Two Variables
Lagrange Multipliers– A tutorial to the introduction of Lagrange multipliers concepts.

Maximum and Minimization– A study guide for those familiar with the maximum and minimization of multiple variables.

Change of Variables for Multiple Integrals
Jacobian for polar and spherical coordinates – Concise instruction on computing this mathematical function from Cartesian to polar coordinates.
Change of Variables– In depth instructive notes on the change of variables.

Vector Fields
Introduction to Vector Fields– Assistant Professor Pete Clark’s handout on the fundamentals of vector fields.
Vector Fields– An applet module displaying vector fields.
Divergence and Curl– A lab using Maple commands for calculating divergence and curl.

Line Integrals
Introduction to Line Integrals– Explanation to line integrals beside plane curves.

Line Integrals of Vector Fields – Guide to line integrals and evaluating line integrals through a given vector field.

Fundamental Theorem for Line Integrals– The basics on line integrals derived from proven theorems.
Green’s Theorem – Mathematical notation for this theorem that is equal to the divergence theorem.

Parametric Surfaces
Parametric Surfaces– Use 3-D graphing to enter functions to produce surfaces in three dimensions.

Parametric Surfaces Tool– Enter formulas to create parametric surfaces.

Surface Integrals
Flux– Explains how to calculate the flux integral over an unlimited surface.

Stokes’ Theorem – Gives examples on computing this surface integral.

Surface Integrals– Defined surface integrals and provides instruction on calculating surface integrals.

Matrices
Introduction to Matrices– A thorough guide with several topics on Matrices.

Determinants– Gives clear history of matrices and determinants and the several solutions substitutions over the years.
Testing for Linear Dependence of Vectors– Provides definitions to notations of vectors and the theorem to testing for linear dependence of vectors.
Systems of Linear Equations –A short review of examples of linear equations.

Gaussian Elimination – Shows how to compute the algorithm for solving matrix formulas.
Eigenvalues and Eigenvectors– Tamara Anthony Carter provides precise essentials to this study of vectors.

Vector calculus isn’t just mathematical formulas related to separation and integration of vector fields. It is an essential part in figuring solutions for the study of matter, as well.