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18.02

Calculus

5-0-7 units. CALC II · Undergraduate

Calculus of several variables. Vector algebra in 3-space, determinants, matrices. Vector-valued functions of one variable, space motion. Scalar functions of several variables: partial differentiation, gradient, optimization techniques. Double integrals and line integrals in the plane; exact differentials and conservative fields; Green's theorem and applications, triple integrals, line and surface integrals in space, Divergence theorem, Stokes' theorem; applications.

Fall 2007 · Multivariable Calculus

Topics

  • Vectors
  • Dot product
  • Determinants; cross product
  • Matrices; inverse matrices
  • Square systems; equations of planes
  • Parametric equations for lines and curves
  • Level curves; partial derivatives; tangent plane approximation
  • Max-min problems; least squares
  • Second derivative test; boundaries and infinity
  • Differentials; chain rule
  • Gradient; directional derivative; tangent plane
  • Lagrange multipliers
  • Non-independent variables
  • Partial differential equations; review
  • Double integrals
  • Double integrals in polar coordinates; applications
  • Change of variables
  • Vector fields and line integrals in the plane
  • Path independence and conservative fields
  • Gradient fields and potential functions
  • Green’s theorem
  • Flux; normal form of Green’s theorem
  • Simply connected regions; review
  • Triple integrals in rectangular and cylindrical coordinates
  • Spherical coordinates; surface area
  • Vector fields in 3D; surface integrals and flux
  • Divergence theorem
  • Divergence theorem (cont.): applications and proof
  • Line integrals in space, curl, exactness and potentials
  • Stokes’ theorem
  • Stokes’ theorem (cont.); review
  • Final review
  • Final review (cont.)

Readings

  • “Text” refers to the course textbook: Edwards, Henry C., and David E. Penney. Multivariable Calculus . 6th ed. Lebanon, IN: Prentice Hall, 2002. ISBN: 9780130339676.
    Listed reading
  • LEC #: 0 · TOPICS: Vectors · READINGS: Text: Section 12.1
    Reading assignment
  • LEC #: 1 · TOPICS: Dot product · READINGS: Text: Section 12.2
    Reading assignment
  • LEC #: 3 · TOPICS: Matrices; inverse matrices · READINGS: Notes: Sections M.1 and M.2
    Reading assignment
  • LEC #: 5 · TOPICS: Parametric equations for lines and curves · READINGS: Text: Sections 12.4 and 10.4
    Reading assignment
  • LEC #: 11 · TOPICS: Differentials; chain rule · READINGS: Text: Sections 13.6-13.7
    Reading assignment
  • LEC #: 12 · TOPICS: Gradient; directional derivative; tangent plane · READINGS: Text: Section 13.8
    Reading assignment
  • LEC #: 13 · TOPICS: Lagrange multipliers · READINGS: Text: Section 13.9, through page 922
    Reading assignment
  • LEC #: 14 · TOPICS: Non-independent variables · READINGS: Notes: Section N
    Reading assignment
  • LEC #: 15 · TOPICS: Partial differential equations; review · READINGS: Notes: Section P
    Reading assignment
  • LEC #: 20 · TOPICS: Path independence and conservative fields · READINGS: Text: Section 15.3
    Reading assignment
  • LEC #: 21 · TOPICS: Gradient fields and potential functions · READINGS: Notes: Section V2
    Reading assignment
  • LEC #: 22 · TOPICS: Green’s theorem · READINGS: Text: Section 15.4
    Reading assignment
  • LEC #: 23 · TOPICS: Flux; normal form of Green’s theorem · READINGS: Notes: Sections V3 and V4
    Reading assignment
  • LEC #: 24 · TOPICS: Simply connected regions; review · READINGS: Notes: Section V5
    Reading assignment
  • LEC #: 27 · TOPICS: Vector fields in 3D; surface integrals and flux · READINGS: Notes: Sections V8 and V9
    Reading assignment
  • Edwards, Henry C., and David E. Penney. Multivariable Calculus . 6th ed. Lebanon, IN: Prentice Hall, 2002. ISBN: 9780130339676.
    Listed reading
Spring 2006 · Multivariable Calculus

Topics

  • Vectors in 2- and 3-space
  • Determinants of Orders 2 and 3
  • Matrices; Inverse Matrices
  • Solving Systems of Linear Equations; Lines, Planes
  • Parametric Curves; Velocity, Acceleration
  • Kepler’s Second Law
  • Level Curves, Partial Derivatives, Tangent Plane
  • Max-Min Problems
  • 2nd Derivative Test; Boundaries and Infinity
  • Differentials; Chain Rule
  • Gradient, Directional Derivative
  • Lagrange Multipliers
  • Non-independent Variables
  • Partial Differential Equations
  • Double and Iterated Integrals
  • Double Integrals in Polar Coordinates
  • Change of Variables
  • Triple Integrals in Rectangular and Cylindrical Coordinates
  • Spherical Coordinates
  • Line Integrals in the Plane
  • Gradient Fields and Path Independence
  • Conservative Fields and Potential Functions
  • Green’s Theorem
  • Simply-connected Regions
  • Flux Form of Green’s Theorem
  • Vector Fields in 3-space; Surface Integrals and Flux
  • Divergence (= Gauss’s) Theorem
  • Divergence Theorem (cont.)
  • Line Integrals in Space, Exactness, and Potentials
  • Stokes’ Theorem
  • Understanding Curl
  • Topological Issues
  • Conservation Laws; Heat/Diffusion Equation
  • Course Review
  • Course Evaluation

Readings

  • “Text” refers to the course textbook: Simmons, George F. Calculus with Analytic Geometry . 2nd ed. New York, NY: McGraw-Hill, 1995. ISBN: 0070576424.
    Listed reading
  • Lec #: 5 · topics: Parametric Curves; Velocity, Acceleration · readings: Text: Sections 18.4, 17.1, 17.4
    Reading assignment
  • Lec #: 6 · topics: Kepler’s Second Law · readings: Text: 17.7 Notes: Section K
    Reading assignment
  • Lec #: 7 · topics: Level Curves, Partial Derivatives, Tangent Plane · readings: Text: Sections 19.1-19.3 Notes: Section TA
    Reading assignment
  • Lec #: 10 · topics: Differentials; Chain Rule · readings: Text: Section 19.6
    Reading assignment
  • Lec #: 11 · topics: Gradient, Directional Derivative · readings: Text: Section 19.5
    Reading assignment
  • Lec #: 12 · topics: Lagrange Multipliers · readings: Text: Section 19.8
    Reading assignment
  • Lec #: 15 · topics: Double and Iterated Integrals · readings: Text: Sections 20.1, 20.2 Notes: Section I.1
    Reading assignment
  • Lec #: 17 · topics: Change of Variables · readings: Text: Section 20.3
    Reading assignment
  • Lec #: 18 · topics: Triple Integrals in Rectangular and Cylindrical Coordinates · readings: Text: Sections 20.5, 10.6
    Reading assignment
  • Lec #: 20 · topics: Line Integrals in the Plane · readings: Text: Section 21.1 Notes: Section V1
    Reading assignment
  • Lec #: 21 · topics: Gradient Fields and Path Independence · readings: Text: Section 21.2 Notes: Section V2.1
    Reading assignment
  • Lec #: 27 · topics: Divergence (= Gauss’s) Theorem · readings: Text: Section 21.4 Notes: Section V10
    Reading assignment
  • Lec #: 30 · topics: Stokes’ Theorem · readings: Text: Section 21.5 Notes: Section V4.3, V13
    Reading assignment
  • Simmons, George F. Calculus with Analytic Geometry. 2nd ed. New York, NY: McGraw-Hill, 1995. ISBN: 0070576424.
    Listed reading

Prerequisites & course information

Prereq: Calculus I (GIR) U (Fall, Spring) 5-0-7 units. CALC II Credit cannot also be received for 18.022 , 18.02A , CC.1802 , ES.1802 , ES.182A

Sources