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Courses: Calculus 3

Multivariate Calculus Overview

Calculus is the study of things in motion. This course builds upon the foundations of calculus as we explore the third dimension. The goal of this course (and the calculus sequence as a whole) is to develop problem-solving skills and to train students to perform mathematical operations. Calculus 3 is the study of motion in the third dimension. To succeed in this course, it is necessary to have a good understanding of the fundamentals of calculus topics discussed in Calculus 1 and Calculus 2.

 

By the end of the course, you will become more proficient with various analytical and computational techniques. You learn topics such as vector-valued functions, multivariate differentiation, and multivariate integration strategies. We will discuss different coordinate systems and how to use them for multivariate differentiation and integration. Students will gain an adequate understanding of the main components of vector calculus.

Course Description: 

Calculus is the study of things in motion. This course builds upon the foundations of calculus as we explore the third dimension. The goal of this course (and the calculus sequence as a whole) is to develop problem-solving skills and to train students to perform mathematical operations. Calculus 3 is the study of motion in the third dimension. To succeed in this course, it is necessary to have a good understanding of the fundamentals of calculus topics discussed in Calculus 1 and Calculus 2.

 

 

By the end of the course, you will become more proficient with various analytical and computational techniques. You learn topics such as vector-valued functions, multivariate differentiation, and multivariate integration strategies. We will discuss different coordinate systems and how to use them for multivariate differentiation and integration. Students will gain an adequate understanding of the main components of vector calculus.

 

Prerequisite: Calculus 1 or AP Calculus AB.

Course Topics

  • Vectors
    • 2-Dimensional
    • 3-Dimensional
    • Dot Product
    • Cross Product
  • Equations of Lines
  • Equations of Planes
  • Quadric Surfaces
  • Space Curves
  • Parameter Equations and
    Vector-Valued Functions
  • Derivatives and Integrals
    • Vector Differentiation Rules
  • Arc Length
  • Curvature
  • Motion in Space
    • Velocity and Acceleration
    • Tangential and Normal Components
    • Newton’s Second Law of Motion
    • Kepler’s Laws
  • Multivariate Functions
    • Domain and Range
    • Level Curves
    • Contour Maps
    • Level Surfaces
  • 3-Dimensional Coordinate Systems
    • Cartesian Coordinates
    • Cylindrical Coordinates
    • Spherical Coordinates
  • Limits and Continuity
  • Partial Derivatives
  • Tangent Planes
  • Multivariate Chain Rule
  • Implicit Differentiation
  • Direction Derivatives
  • The Gradient Vector
  • Optimization
    • Second Derivative Test
    • Lagrange Multiplers
  •  
  • Multivariate Riemann Sums
  • Double Integral
    • Iterated Integrals
    • General Regions
    • Polar Coordinates
    • Area and Surface Area
    •  Applications
      • Density and Mass
      • Moment of Inertia
      • Probability
  • Triple Integrals
    • Iterated Integrals
    • General Regions
    • Cylindrical Coordinates
    • Spherical Coordinates
    • Volume
  • Jacobian 
  • Vector Fields
  • Line Integrals
    • Path Independence
    • Conservative Vector Fields
    • Fundament Theorem of Line Integrals
  • Green’s Theorem
  • Curl and Divergence
  • Parametric Surfaces
    • Tangent Plane
    • Surface Area
  • Surface Integral
  • Flux Integral
  • Stoke’s Theorem
  • The Divergence Theorem

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