8:00 -- 8:50 DBH 1600
DISCUSSION: Tu-Th 1:00 -- 1:50 HICF 100M (MAP:BLDG # 523)
Tu-Th 5:00 -- 5:50 ICF 103 (MAP:BLDG # 315)
INSTRUCTOR: Martin Zeman, RH 410E
OFFICE HOURS: Wednesday 10:00am - 12:00noon and by appointment
TA: Raymond Watkin, RH 410R
OFFICE HOURS: Tuesday 2:00pm - 3:00pm and Thursday 2:00pm - 4:50pm
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MIDTERM 1 INFORMATION MIDTERM 2 INFORMATION FINAL EXAM INFORMATION
WEEK 10 M: Orientation of a surface. Surface integral of a vector field. Flux of a vector field across a surface. Heat flow. Positive orientation of a boundary of an oriented surface. Stokes theorem. curl(F)=0 implies F is conservative. Recommended practice problems: Section 16.7 Exercise 21-32, 47, 48 and Section 16.8 Exercise 7-10, 11(a), 12(a), 17
M: Curl and divergence. Operator ``nabla". Gradient as the multiple of ``nabla" with scalar a field, curl as the cross product of ``nable" with a vector field, and divergence as the scalar product of ``nabla" with a vector field. Criterion: curl(F)=0 iff F is conservative. Theorem: div(curl(F))=0. Recommended practice problems: Section 16.5 Exercise 1-8, 9-11, 12, 19, 20, 21, 22
W: Parametric surfaces. Examples: Cylinder, sphere. Partial derivatives of r(u,v) and tangent planes. Normal vector to the tangent plane as the cross product r_u(u,v) x r_v(u,v). Parametric and non-parametric equations of the tangent plane. Recommended practice problems: Section 16.6 Exercise 1-2, 3-6, 13-18, 19-26, 33-36
F: Smooth/piecewise smooth surfaces. Surface integral of a function over a surface. Example: Sphere with radius a, f(x,y,z)=y^2. Orientation of a surface. Recommended practice problems: Section 16.7 Exercise 5-20, 39, 40
M: Applications of Green's theorem: Area surrounded by a closed curve. Discussion of the two assumptions in Green's theorem: Simplicity of C, positive orientation. Extended version of Green's theorem: The vector field can be defined on any open simply connected region. Applications of Green's theorem to vector fields defined on open connected regions with holes. Recommended practice problems: Section 16.4 Exercise 11-14, 19, 27, 28, 29
W: Green's theorem and regions which are not simply conncted. Explanation, why Green's theorem holds. Turbulence. Midterm review. Sample writing for midterm
F: Midterm 2
W: Geometry of plane and space: closed curve, simple curve, open region, simply connected region. Criteria on conservativity of a vector field: Euivalences of the following:
(A) Path independence of the integral of Fdr versus the integral of Fdr along a closed curve is 0.
(B) Path independence of the integral of Fdr versus F being conservative.
(C) Equality of partial derivatives Q_y=P_x versus P(x,y)i+Q(x,y)j being conservartive.
Recommended practice problems: Section 16.3 Exercise 3-10, 12-18, 19-20, 25-26, 31-34, 35, 36
F: Orientation of a closed curve. Green's theorem. Applications of Green's theorem: Evaluating line integral, calculating the area of a region surrounded by a close curve. Recommended practice problems: Section 16.4 Exercise 1-4, 5-10, 17, 18