Math 13, Multivariable calculus, Course Summary
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24 lectures + 1 bonus lecture.
1. Vectors and vector products. Vector valued functions, limits.
2. Differentiation of vector valued functions, velocity, smoothness,
properties of the derivative.
3. Arc length, speed, unit tangent, curvature, principal normal.
4. Tangential and normal components of acceleration. Radius of curvature.
Functions of several variables, graphs, level curves, level
surfaces. Limits and paths of approach.
5. Properties of limits. Continuity. Partial derivatives. Equality of
second partial derivatives. Higher order derivatives.
6. Geometric interpretation of partial derivatives. Equation of tangent
plane. Linear approximation, differentials and differentiability.
7. The chain rule: 1 independent variable, 2 independent variables, general
case, implicit differentiation.
8. The gradient vector: 2 variable case, steepest ascent, directional
derivatives, gradient perpendicular to level curves. 3 variable case,
gradient perpendicular to level surfaces, equation of tangent line/plane.
9. Maxima and minima, critical points, saddle points, second derivatives
test.
10. Maxima and minima in 3 or more variables, critical points, second
derivatives test, completing the square, checking the boundary.
Constrained maxima/minima, Lagrange multipliers.
11. Lagrange multipliers in 3 or more variables.
12. Definition of double integrals, calculation of volumes, iterated
integrals.
13. Iterated integrals, type I and type II regions, changing the order of
integration. Double integrals in Polar coordinates.
14. Polar coordinates, general change of variables formula for double
integrals, Jacobians, formula for the area element.
15. Change of variables example.
Triple integrals, definition, iterated integrals, regions of different
types. The six possible orders of integration.
16. Change of variables in triple integrals. Formula for the volume
element. Cylindrical coordinates. Spherical coordinates.
Vector calculus: vector fields.
17. Conservative vector fields. Conservation of energy. Line integrals.
Physical examples of vector fields. Physical interpretation of line
integrals. Properties of line integrals. Example of line integrals that are
not path independent.
18. Conservative vector fields and path independence. Method for finding
the potential of a conservative vector field. Differential condition for
conservative vector fields on simply connected regions in the plane.
Parametric surfaces, examples, graphs, surfaces of revolution.
19. Surface area formula. Finding the surface element. Surface area of a
sphere. Surface area of a graph. Scalar surface integrals. Orientation of
surfaces, orientable and nonorientable surfaces. Vector surface integrals.
The surface integral of the point charge field over an enclosing sphere.
20. Interpretation of vector surface integrals. The flux through a surface.
Finding the unit normal. Finding the vector surface element. Calculation of
vector surface integrals.
21. Divergence and Curl. Definition and basic properties. Interpretation of
divergence. Flow per unit volume. Interpretation of curl. Circulation per
unit normal surface area. The right hand rule.
22. Advanced properties of divergence and curl. curl grad = 0, div curl = 0,
and the converse statements.
Stokes theorem. The right hand rule. Orienting the boundary correctly.
Using Stokes theorem to calculate surface integrals.
23. Proof that an irrotational vector field on a simply connected region is
conservative. Proof of Stokes theorem. Cancellation of internal
boundaries. Surfaces with no boundary.
The Divergence theorem, correct orientation of the boundary. Using the
divergence theorem to compute surface integrals.
24. Divergence theorem examples. Independence of surface integrals on the
surface when the divergence is zero. The flux of a point charge through an
enclosing surface. The divergence of the point charge field. Cancellation
of internal boundaries. Proof of the divergence theorem.
25. Optional bonus lecture: Orthogonal curvilinear coordinates. Formula of
the line and volume elements. Formula for gradient, divergence and curl in
terms of the line element. Cylindrical coordinates. Spherical coordinates.