- Extended Math Learning Center hours for finals
- Opening 9:30 (instead of 10:30) AM weekdays
- 7:00 - 10:00 PM Sunday, May 8
- Final
- Monday, May 9, 12:00 noon
- In our classroom
- Comprehensive, but emphasizing material since 2nd hour exam (e.g., multiple integrals, line integrals, Green’s theorem, applications, etc.)
- Designed for 2 hours, you’ll have 3+
- Rules and format otherwise similar to hour exams
- Donuts and cider
- SOFIs
- Helpful to me in planning future offerings of this course
- Helpful to me in planning future offerings of this course
- Section 16.5, introduction and “Parameterizations of Surfaces”
- Position of point on surface as parametric function of 2 variables
- Can find area as double integral
- r(u,v) = f(u,v)i + g(u,v)j + h(u,v)k = ⟨ x, y, z ⟩
- Plotting with muPad
plot::Surfacefunction
- Examples
- Book’s x2 + (y-3)2 = 9 cylinder
- Compare in muPad to book’s form
![cyl := matrix(1,3,[3*cos(u),3*sin(u),v]), p := plot::Surface(cyl,u=0..2*PI,v=0..5), plot(p) draws cylinder](mpparcylinder.png)
- Compare in muPad to book’s form
- Book’s z = √(x2+y2) cone
- y = x2 parameterize as ⟨ t, t2 ⟩
- Do something similar here:
- ⟨ u, v, √(u2+v2) ⟩,
-1≤u≤1, -1≤v≤1
![c1 := matrix(1,3,[u,v,sqrt(u^2+v^2)]), plot(plot::Surface(c1,u=-1..1,v=-1..1)) draws cone](mpparcone.png)
- ⟨ u, v, √(u2+v2) ⟩,
-1≤u≤1, -1≤v≤1
- Cylinder z = x3
- ⟨ u, v, u3 ⟩
- This is a mathematician’s cylinder, i.e., a curve extruded along a line
![cyl2 := matrix(1,3,[u,v,u^3]), plot( plot::Surface(cyl2,u=-1..1,v=-1..1) ) draws sheet with cubic curve to it](mpparcube.png)
- Book’s x2 + (y-3)2 = 9 cylinder
- Derivatives and tangents
- Divergence and curl
- Read
- “Divergence in Three Dimensions” from section 16.8 (starting on page 1015)
- “The Curl Vector Field” from section 16.7 (starting on page 1002)