Dave's Web Corner

MTH-261 Week 15 (December 1-7)

• Outline
• Assessments
• Handouts
• Video
• Technology

Outline

16.8 Stokes' Theorem

• recommended exercises
  • 3, 5, 7, 9, 11, 13, 15, 17, 18
• learning objectives
  • explain the meaning of Stokes’ theorem
  • use Stokes’ theorem to evaluate a line integral
  • use Stokes’ theorem to calculate a surface integral
  • use Stokes’ theorem to calculate a curl
• additional notes
  • the first extension of green's theorem to space
  • the circulation of a field along a closed path in space
    • \( \displaystyle{ \oint_{C} \vec{F}\cdot d\vec{r} = \int\int_{S} \left(\nabla\times\vec{F}\right)\cdot\vec{N}\,dS }\)
  • circulation density

16.9 The Divergence Theorem

• recommended exercises
  • 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 24
• learning objectives
  • explain the meaning of the divergence theorem
  • use the divergence theorem to calculate the flux of a vector field
  • apply the divergence theorem to an electrostatic fields
• additional notes
  • the second extension of green's theorem to space
  • flux across closed, oriented surfaces
    • \( \displaystyle{ \int\int_{S} \vec{F}\cdot\vec{N}\,dS = \int\int\int_{G} \nabla\cdot\vec{F}\,dV }\)
  • sources and sinks, flux density

Section 16.10 Summary

• The Fundamental Theorems of Calculus
  • the Fundamental Theorem of Calculus
    • \( \displaystyle{ \int_{a}^{b}f'(x)\,dx=f(b)-f(a) } \)
  • the Fundamental Theorem of Line Integrals
    • \( \displaystyle{ \int_{C}\nabla f\cdot d\vec{r} = f\left(\vec{r}(b)\right)-f\left(\vec{r}(a)\right) } \)
  • Green's Theorem (Curl and Divergence forms)
    • \( \displaystyle{\int\int_{R} \left( \frac{\partial g}{\partial x}-\frac{\partial f}{\partial y}\right)\,dA = \oint_{C} \vec{F}\cdot\vec{T}\,ds} \)
    • \( \displaystyle{\int\int_{R} \left( \frac{\partial f}{\partial x}+\frac{\partial g}{\partial y}\right)\,dA = \oint_{C} \vec{F}\cdot\vec{N}\,ds} \)
  • Stokes' Theorem
    • \( \displaystyle{\int\int_{\Sigma} \left(\nabla \times \vec{F}\right) \cdot \vec{N}\,dS = \oint_{C} \vec{F}\cdot\vec{T}\,ds } \)
  • Gauss' Theorem
    • \( \displaystyle{ \int\int\int_{G} \nabla\cdot\vec{F}\,dV = \int\int_{S} \vec{F}\cdot \vec{N}\,dS } \)
• Uniting the Theories of Electricity and Magnetism
  • Maxwell's Equations
    • Gauss' Law of Electricity
      • \( \displaystyle{ \epsilon_{0}\int\int_{\Sigma}\vec{E}\cdot d\vec{S} = q } \)
    • Gauss' Law of Magnetism
      • \( \displaystyle{ \int\int_{\Sigma}\vec{B}\cdot d\vec{S} = 0 } \)
    • Faraday's Law of Induction
      • \( \displaystyle{\int_{C} \vec{E}\cdot d\vec{r} = -\mu_{0}\frac{d}{dt}\int\int_{\Sigma} \vec{B}\cdot d\vec{S} }\)
    • Ampere's Law (as extended by Maxwell)
      • \( \displaystyle{\int_{C} \vec{B}\cdot d\vec{r} = \mu_{0}I+\mu_{0}\epsilon_{0}\frac{d}{dt}\int\int_{\Sigma} \vec{E}\cdot d\vec{S}} \)

Assessments

• Deadlines
  • All homework is due at the beginning of class on the date indicated.
  • All exams are given in class on the date indicated.
  • Late homework is not accepted and there are no make-ups for missed exams. Contact me before the deadline if you have a conflict with a deadline and we will find a mutually agreeable solution.
  • Solutions are posted under these links after the deadline.
• Homework
  • due Tuesday, December 3:
    • 16.6 - , 16.7 -
  • due Tuesday, December 10:
    • 16.8 - , 16.9 - , 16.10 -
• Exams
  • Thursday, December 12: Exam 3
    • Exam 3 (solutions)
    • Exam 3 covers sections 15.8-15.10 and 16.1-16.10

Handouts

• Announcements
  • Using Mathematica.pdf
  • Schmidt Award
  • Math Graduate Award
  • Honors Program introduction
  • Tutoring Center services
• Formula Sheets
  • Formulas 1
  • Formulas 2
  • Formulas 3
  • Formulas 4
• Additional Exam Practice
  • Sample Exam Problems.zip
  • Other Exam Problems.zip
  • Multiple Integral Problems.zip
  • Green Gauss Stokes Problems.zip
• Cylindrical and Spherical Coordinates
  • The Polar Cap
• Earlier Calculus Facts
  • The Hyperbolic Functions
  • Reduction Formulas
  • Wallis' Formula
  • Center of Mass
• Green's, Gauss', and Stokes' Theorems
  • Circle Parameterizations
  • Conservative Fields
  • Green's Theorem (Curl Form)
  • Green's Theorem (Divergence Form)
  • Surface Integral Example
  • Gauss' Theorem
  • Stokes' Theorem

Video

• Recommended Problems
  • Chapter 6
  • all MTH-261 Recommended Problems
• Topics
  • The Intersection of Two Planes
  • Arc Length Parameterization
  • Frenet Frame Movie
  • An Interesting Curve in Space
  • Optimization Problem Comparison
  • all MTH-261 Topics

Technology

• Mathematica
  • Ice Cream Cone.nb
  • Density of a Plate.nb
  • Center of Mass Moments of Inertia.nb
  • 261-exercise-06-02-80-alt-demo
  • Flux through Cone