Dave's Web Corner

MTH-261 Week 12 (November 10-16)

recommended exercises
- 5, 7, 9, 11, 15, 17, 19, 23, 25, 27
learning objectives
- determine the image of a region under a given transformation of variables
- compute the Jacobian of a given transformation
- evaluate a double integral using a change of variables
- esvaluate a triple integral using a change of variables
additional notes
- when a polar, cylindrical, or spherical conversion just won't do
- clues for conversion
  - patterns in the limits or the integrand
- the Jacobian is (multiple choice)
  - a wrestling hold
  - a John Wayne movie
  - the price of conversion
- the truth about \(u\)-substitution

recommended exercises
- 4, 5, 6, 11-14, 15-18, 19, 21, 23, 24, 25, 27, 29-32, 33
learning objectives
- recognize a vector field in a plane or in space
- sketch a vector field from a given equation
- identify a conservative field and its associated potential function
additional notes
- a vector field is a vector-valued function of a vector variable
  - \(\vec{F}(x,y)=\langle f(x,y),g(x,y)\rangle\)
  - \(\vec{F}(x,y,z)=\langle f(x,y,z),g(x,y,z),h(x,y,z)\rangle\)
- natural vector fields and their creation
  - gravitational fields
  - electromagnetic fields
- the gradient field
  - always normal to level curves
  - conservative vector fields
  - potential functions

recommended exercises
- 3, 5, 9, 13, 16, 17, 21, 23, 25, 29, 33, 39, 41, 43, 45
learning objectives
- calculate a scalar line integral along a curve
- calculate a vector line integral along an oriented curve in space
- use a line integral to compute the work done in moving an object along a curve in a vector field
- describe the flux and circulation of a vector field
additional notes
- the contribution of a function to a curve
- alternate notation for line integrals
  - \(\int_{C}f\,ds\), \(\int_{C}f\,dx + g\,dy\,\,(+ h\,dz)\), \(\int_{C}\vec{F} \cdot d\vec{r}\)
- contributions of fields to paths
  - flow: the contribution of a field along a path
    - \(\int_{C}\vec{F} \cdot \vec{T}\,ds\)
    - work: \(\int_{C}\vec{F} \cdot d\vec{r}\)
    - circulation: \(\oint_{C}\vec{F} \cdot d\vec{r}\)
  - flux: the contribution of a field normal to a path
    - \(\int_{C}\vec{F} \cdot \vec{n}\,ds\)

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 Thursday, November 14:
  - (none)
- due Thursday, November 21:
  - 15.10 - , 16.1 - , 16.2 -
- due Tuesday, November 26:
  - 16.3 - , 16.4 - , 16.5 -
Exams
- Thursday, December 12: Exam 3
  - Exam 3 (solutions)
  - Exam 3 covers sections 15.8-15.10 and 16.1-16.10