MATH 2242 (Calc IV / Vector Calculus) Section 090 - Spring 2010 - UNC Charlotte

Class meets: Tuesdays and Thursdays 6:30pm-7:45pm, Fretwell 305.
Final Exam: Tuesday May 11, 8:00-10:30pm, Fretwell 305.
My Email:
My Office: Fretwell 370E. Enter main doors to the math department on the third floor of Fretwell, then turn left by the front desk. My office is down the hallway on the left. Refer to the picture on the right.
My Office Phone: 704-687-3874
My Office Hours: Tuesdays and Thursdays, 5:00pm-6:30pm; also after class, if necessary.

Class Handouts

Syllabus
Midterm Study Guide
Midterm Key
Examples of double integrals
Key to Final Exam Extra Credit
Final Exam Study Guide
Final Exam

Quizzes

Quiz 1, Key
Quiz 2, Key
Quiz 3, Key
Quiz 4, Key
Quiz 5, Key
Quiz 6, Key
Quiz 7, Key
Quiz 8, Key
Quiz 9, Key
Quiz 10, Key

Lecture Notes

Lecture 1: Introduction. Algebraic and geometric vectors. Equations of lines and planes. Dot products, orthogonal projections, and angle between vectors. Matrices, determinants, cross products.

Lecture 2: Properties of the cross product. Cylindrical and spherical coordinates.

Lecture 3: Euclidean space R^n. Cauchy-Schwarz and Triangle inequalities. Matrix multiplication, identities, and inverses. Geometry of real-valued functions. Level sets and sections. Limits and continuity, open disks and open sets in R^n.

Lecture 4: Properties of limits. Differentiation of multi-variable functions. Linear approximations and tangent planes.

Lecture 5: Summary so far. Theorem on differentiation, partial derivatives, and continuity. More tangent planes. Paths and curves, velocity, and tangent lines.

Lecture 6: Properties of the derivative. The multi-variable chain rule. First special case of the chain rule. Gradients and directional derivatives. Geometrical interpretation of the gradient.

Lecture 7: Gradient as a normal to level surfaces. Theorem on equality of mixed partials.

Lecture 8: Acceleration and Newton's second law. Uniform circular motion. Arc length formula and differential of arc length.

Lecture 9: Scalar and vector fields. Gradient vector fields. Conservation of energy. Flow lines.

Lecture 10: The "del" operator. The gradient, divergence, curl, scalar curl, Laplacian, and their physical interpretation.

Lecture 11: Double integrals, Cavalieri's principle, and iterated integrals. The Riemann definition of a double integral. Properties of double integrals. Fubini's theorem.

Lecture 12: The double integral over more general regions. X-simple, y-simple, and simple regions. Reduction of double integrals to iterated integrals. Changing the order of integral. Mean value theorem for integrals. Triple integrals.

Lecture 13: Examples of double integrals, changing the order of integration, and triple integrals. The geometry of maps from R^2 to R^2. One-to-one and onto functions. Linear maps.

Lecture 14: The change of variables theorem for multiple integrals. Jacobian determinants. The Gaussian integral. The change of variables formulas for polar, cylindrical, and spherical coordinates. Applications of multiple integrals: average values, center-of-mass, and moments of inertia.

Lecture 15: Examples of maps from R^2 to R^2 and change-of-variables. Path integrals of scalar functions.

Lecture 16: Examples of path integrals. Line integrals of vector fields and applications to physics (e.g. work). The unit tangent vector. Differential forms notation for line integrals. Reparameterizations of line integrals.

Lecture 17: Examples of line integrals and reparameterizations. Orientation of line integrals. The "opposite" path. The fundamental theorem of calculus for gradient vector fields. Line integrals over geometric curves.

Lecture 18: Parameterized surfaces. Tangent vectors and normal vectors to a surface. Regular surfaces. Tangent plane to a regular point of a parameterized surface.

Lecture 19: Examples of parameterized surfaces. Area of a parameterized surface. The surface area of a graph.

Lecture 20: Surface integrals of scalar and vector fields. Interpretation of the latter as flux integrals. Orientation of surfaces and its affect on surface integrals. Reparameterizations of a surface.

Lecture 21: Green's theorem (really Stokes' theorem in the plane). Area via Green's theorem. Relationship between Green's theorem and the scalar curl of a vector field on R^2. The unit normal vector to a parameterized path. Divergence theorem in the plane.

Lecture 22: Examples of Green's theorem. Stokes' theorem in R^3. Independence of surface for Stokes' theorem. Some examples.

Lecture 23: Stokes' Theorem example. Conservative vector fields. Big theorem: The following are equivalent for a vector field F: (a) path independence of line integrals, (b) F is conservative, (c) F is curl-free, and (d) F is a gradient field. Some examples.

Lecture 24: Examples of conservative fields. Gauss' Divergence Theorem. Examples. The relationship between integral and differential forms of Maxwell's equations from physics.

Homework

Daily homework #1: Read 1.1-1.3; 1.1/1-5 odd, 9-21 odd, 26*; 1.2/1-23 odd
Daily homework #2: Read 1.3-1.5 and 2.1; 1.3/1-5 all, 7-15 odd, 17*, 19, 21*, 25, 28* (for 5 and 7, see the boxes on pp.48-49)
Daily homework #3: Read 2.1-2.2; 1.4/1-5 all, 6*, 7*, 9-15 odd. 1.5/1, 2, 3-13 odd, 15*, 17, 19.
Daily homework #4: 2.1/1, 2, 5-17 odd, 19*-29* odd.
Daily homework #5: Read 2.3, 2.4; 2.3/1, 3, 7, 8, 11. Also, study example 4 on pp.130-131 carefully.
Daily homework #6: Read 2.4, 2.5; 2.3/5, 6c, 9, 10, 12, 13, 15, 17, 20*.
Daily homework #7: Read 2.5, 2.6; 2.4/1-15 odd, 17*, 19*.
Daily homework #8: Read 2.6, 3.1; 2.5/1, 2cdef, 3, 4, 5abc, 7, 8*, 9-12, 17*, 20*.
Daily homework #9: Read 4.1, 4.2; 2.6/1, 2acd, 3ab, 4ac, 5ab, 6ab, 8, 13ab, 15*, 16. (For #5 and #8, (re)read p.166-170.)
Daily homework #10: Read 4.2, 4.3; 3.1/1-3, 9. 4.1/1-9, 11, 14, 17, 20*, 21*.
Daily homework #11: Read 4.3, 4.4; 4.2/1-9 odd, 11*, 12, 13*-19*.
Daily homework #12: Read 4.4; 4.3/1-15 odd, 18*.
Daily homework #13: Read 5.1-5.2; 4.4/1-31 odd.
Daily homework #14: Read 5.2-5.3; 5.1/1-11 odd.
Daily homework #15: Read 5.4-5.5; 5.2/1-7 odd, 8*, 9*, 11*; 5.3/1-4, 6-8, 11, 12, 14.
Daily homework #16: Read 6.1-6.2; 5.4/1-2, 4-5, 10, 12, 13, 14*, 15*; 5.5/1-8, 9, 12*, 13-25 odd, 27*, 29*.
Daily homework #17: Read 6.1-7.1; 6.1/1-7, 8*, 9*, 10, 11*, 12; 6.2/1-6, 7-13 odd, 17-21 odd, 26-27, 29, 31.
Daily homework #18: Read 6.2-7.1; 6.2/1-6, 7-13 odd, 17-21 odd, 26-27, 29, 31.
Daily homework #19: Read 7.1-7.5; 7.1/1-7 odd, 6*, 11, 12, 14; 7.2/1-7 odd, 4*, 9-15 odd, 17*
Daily homework #20: Read 7.3-7.5; 7.3/1-11 odd, 14, 16*, 17ab.
Daily homework #21: Read 7.5-7.6; 7.4/1, 2, 3, 5-13 odd, 6*, 10, 22.
Daily homework #22: Read 7.5, 7.6, 8.1; 7.5/1*, 2, 3, 5, 7, 9*, 11, 15; 7.6/1-9 odd, 10, 11-17 odd.
Daily homework #23: Read 8.1-8.2; 8.1/1-5, 7-15 odd, 6*, 8*, 10*, 12*, 16*, 18*, 20*.
Daily homework #23: Read 8.2-8.3; 8.2/3-11 odd, 8, 21, 14*, 16*.
Daily homework #24: Read 8.3-8.4; 8.3/1-5, 7-9, 12, 13, 16*, 18*. [Hint: For 12a, switch to polar coordinates.]
Daily homework #25: Read 8.4; 8.4/1-10, 11*, 15*, 18*.

Extra Credit

Midterm extra credit: Linear approximation problems from chapter 2.
Final exam extra credit: 6.3/2, 4, 5, 8. Also, find the average value of f(x,y)=x^2+y^2 over the solid cone x^2+y^2 <=z^2 between z=0 and z=2.