**Multivariable Calculus (Math 3354)**

Rick Wicklin

Rick's Office: 451 Vincent Hall, 626-8308 or 626-1324

`fjw@geom.umn.edu, http://www.geom.umn.edu/~math335x`

Textbook: *Multivariable Calculus* by Damiano and
Freije, Brooks-Cole, 1996.

Text for sale ($20) at the Special Projects Office, 115 VinH, 8:30-4:00pm.

**TAs:** Bob Hesse (Workshop Czar)

Sec 1 (9:00-9:55): E. Schirmacher

Sec 2 (11:15-12:05): G. Naughton

Sec 3 (13:25-14:15): T. Brule

This course will focus on the geometry of curves and surfaces (and related differential equations) for the first six weeks of the course. The remainder of the course will focus on the local linear approximation of a function or a vector field.

**Week 1**- (9/26-10/3)The Geometry of Parametric Curves
- Read §3.1,3.3 and §2.1-2.3.
- Vectors, parametric lines, tangent vectors to curves,
velocity vectors, chain rule for
*g*(*f*(*t*)) where , . The difference between a graph and an image, and between slope and a tangent vector. - Linear Algebra: Review of basic vector concepts.
- Graded Lab: (Lab 1) Introduction to Maple derivatives, vector notation, plotting planar graphs and images, plotting spacecurves, printing.
- Homework: (HW 1) Families of parametric curves.
- Workshop topics:
- Inventing a parametrization of simple planar and nonplanar curves. Can different parametrizations have the same image? Computing of the velocity vector based at the point of tangency.
- Parametrized curves arising from solution of simple (uncoupled) linear ODEs.

**Weeks 2-3**- (10/4-18) The Geometry of Surfaces in
- Read §1.1-1.2 and §4.2-4.3.
- Graphs of , planes as graphs, coordinate curves, slope in coordinate directions (=partial derivatives), directional derivatives and the gradient. Parametrized surfaces and tangent vectors to coordinate curves on surface.
- Linear Algebra: (NEW!) linear combination (span) of vectors
- Graded Lab: (Lab 3) Exploring the geometry of surface graphs and directional derivatives by examining ``cross-section'' of surfaces.
- Homework: (HW 2) Construct a surface; (HW 3) Numerical estimates of directional derivatives.
- Workshop topics:
- Discussion 4.1
- (Lab 2) Parametrizing planes in Maple
- Discussion 4.3

**GATEWAY I**- 10/17
**EXAM I**- 10/22. Covers weeks 1-3.
**Week 4**- (10/23-29) Max/Min/Saddles
- Read §4.5, 5.1, 5.2
- Critical points and how to find them, types of critical points (saddle versus extremum), gradient vector fields and their equilibria, optimization of scalar functions.
- Linear Algebra: review of orthogonality of two vectors.
- Graded Lab: (Lab 5)
Relationship of a gradient vector field, , to the
graph of
*f*. Discovering linear phase portraits of nodes and saddles. - Homework: (HW 4) Gradients; (HW 5) Discussion 5.2
- Workshop topics:
- Discussion 4.4
- (Lab 4) Intro to Matlab and PPLANE
- Discussion 5.1
- Locating and classifying critical points. Second derivative test.
- Are all vector fields gradient fields? How can we tell geometrically and analytically whether a vector field is gradient?

**Weeks 5-6**- (10/30-11/13) Level Sets
- Read §4.5, 5.3
- Implicit curves, gradient vector fields revisited, Hamiltonian vector fields, implicit differentiation, change of topology across singular levels, Lagrange multipliers.
- Graded Lab: (Lab 7) Planetary motion (Kepler's restricted planar 2- and 3-body problem)
- Homework: (HW 6) Changing topology as levels change and relating that change to critical points on the graph of the function; (HW 7) Lagrange multipliers (letter to boss)
- Workshop topics:
- Relationship of gradient and Hamiltonian vector fields to the geometry of surfaces.
- (Lab 6) The geometry of the gravitational potential function.
- Discussion 5.3: The geometry of Lagrange multipliers.

**GATEWAY II**- 11/12
**EXAM II**- 11/14 Covers weeks 1-6, with an emphasis on weeks 4-6.
**Weeks 7-8.5**- (11/15-11/21) Tangent Planes
- Read §4.1, 4.4
- Multivariable limit, limit along a curve, notions of continuity and differentiability, existence of tangent plane, tangent plane at critical points.
- Linear Algebra: review linear independence, span, linear combinations. matrix-vector multiplication.
- Brainfood:
- Differentiable and non-differentiable surfaces.
- Different ways to compute tangent planes to surfaces: Relationship between tangent planes to graphs, and to parametrized surfaces. Verification of tangent plane to implicitly-defined surfaces.

**Weeks 8.5-10**- (11/22-12/6) Linear Approximations and Jacobians
- Read ??
- Tangent plane as best affine approximation to surface,
Recognition of as a linear map,
Jacobian as best affine approx for vector field at equilibria,
general chain rule:
*D*(*f*(*g*(*x*))=*Dg*(*f*(*x*)).*Df*(*x*) (requires param. surfaces) - Graded Lab: Comparison of trajectories of Hamiltonian vector fields and the vector field obtained by linearization at equilibrium. Relate to geometry of generating function.
- Brainfood:
- Comparison of level sets for nonlinear function and the linear approximation at a critical point.
- Composing a parametrized curve with a parametrization of a surface and using the chain rule to compute derivative information.

**GATEWAY III**- 12/3
**FINAL EXAM**- 12/9, 1:30pm, AndH 350. The final exam covers material
from weeks 1-10 with a slight emphasis on weeks 7-10.

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Last modified: Wed Nov 27 17:23:07 1996

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