M2E: PREVIOUS LECTURES
WEEK 6
M: An example of a non-smooth curve with continuous derivative:
r(t)=(t^2,t^4), -1\le t\le 1. An example of a triple integral in
dimension 3: spiral x(t)= a cos t, y(t)= a sin t, z(t) = at for 0\le t\le 2\pi,
a>0 is a constant. Line integral of a vector field: an introduction and the
integration formula.
Recommended practice problems:
Section 16.2 Exercise 17, 18, 19-22
W: Line integral of a vector field. Integrals according to x,y and z.
Orientation of a curve. Examples (a) F(x,y)=(-y,x), C is given by
r(t)=(1+t^2,t) where -1\le t\le 1, C_1 is the straight line connecting
(2,-1) and (2,1). (b) F(x,y,z) is the gravitational field generated
by an object of mass M placed in the center of the coordinate system. C is
the intersection of the ``cylinder" x^2/a^2+y^2/b^2=z^2 with the plane paralel
with the xy-plane at height c>0 above the xy-plane.
Recommended practice problems:
Section 16.2 Exercise 19-22, 39-44
F: Fundamental theorem for line intergrals. Path independence. Closed
curves. Equivalence of path independence and integral along a closed curve
being 0.
WEEK 5
M: Midterm 1
W: Electric and Gravitational field. Gradient/conservative fields.
Smooth curves in dimension 2. Idea behind a line integral with respect to arc
in dimension 2.
Recommended practice problems:
Section 16.1 Exercises 21-24, 25-25, 29-32, 35, 36
F: Recap on smooth curves. Piecewise smooth curves. Line intergral with
respect to arc length: Integration formula and basic properties, both in
dimension 2 and 3. Applications: length, mass, moments, center of gravity.
Example: upper half-circle with radius a centered in the origin; density
function d(x,y) = a-y.
Recommended practice problems:
Section 16.2 Exercises 1-16, 17, 18
WEEK 4
M: Examples for changes of coordinates in dimension 3: Spherical
coordinates, volume of a torus.
Recommended practice problems:
Section 15.9 Exercises 23-27 and Review for Section~15, Exercises 29-34 on
pages 1062 and 1063
W: Section 16.1 on vector fields. Vector fields in both dimension 2
and 3. Examples of vector fields: F(x,y) = xi,
F(x,y) = -yi + xj, F(x,y) = yi - xj,
gravitational field, electric field.
Recommended practice problems:
Section 16.1 Exercises 1-10, 11-14, 15-18
F: Review
WEEK3
M: Spherical coordinates and the integration formula - recap. Examples
of integration with spherical coordinates: (a) Volume of a ball with radius a,
(b) Mass if the ball from (a) if the density is given by
\sigma(x,y,z)=(x^2+y^2+z^2)e^{{x^2+y^2+z^2}^{5/2}}, (c) volume of the region
surrounded by the cone z=(x^2+y^2)^{1/2} and sphere x^2+y^2+z^2=z: we expressed
these objects in spherical coordinates, the integration will be done next time.
Recommended practice problems:
Section 15.8 Exercises 11-14,15,16,17-18,19-20,21-27
W: Change of coordinates in Dimension 2: Transformations of 2D objects,
Jacobian, integration formula. Example: Polar coordinates.
Recommended practice problems:
Section 15.8 Exercises 29, 30, 31(a), 32(a,b), 33(a), 34, 35, 36, 37(a), 38(a);
Section 15.9 Exercises 1-6 and 7-10
F: Examples on change of coordinates in 2D: Recap of polar coordinates,
change of coordinates via linear transformation in the plane. Change of
coordinates in 3D, the Jacobian and the integration formula. Examples:
cylindrical coordinates, spherical coordinates (spherical to be continued
next time.)
Recommended practice problems:
Section 15.9 Exercises 11-14, 15-19
WEEK 2
M: Holiday.
W: Example: Evaluation of a double integral using polar coordinates.
Cylindrical coordinates and integration using cylindrical coordinates. Example:
Volume of a cone; to be continued.
Recommended practice problems:
Section 15.7 Exercises 13, 15-16, 17-24
F: Examples of integration in cylindrical coordinates: A cone with
height h and radius of the base a; density is K/r. Calculation of volume, mass,
moments and center of gravity. Determining moments of symmetric objects without
actual evaluation of the integral. Spherical coordinates.
Recommended practice problems:
Section 15.7 Exercises 27, 28, 29-30 and Section 15.8 Exercises 1-2, 3-4, 5-6,
7-8, 9-10
WEEK 1
M: General info about the course. Review of double and triple integrals:
The definition of an integral. Integrability. Basic properties of integrals:
linearity, monotonicity, integration over the union of two disjoint areas,
Fubini theorem for rectangles.
Recommended practice problems:
Section 15.1 Exercises 27-34, 35-36 and 37-43.
W: Recap: (A) Types of regions in 2D and 3D. (B) Integration over
regions of various types. (C) Examples of integration over such regions.
Recommended practice problems:
Section 15.2 Exercises 7-10,11-14,15-16,17-22,23-32,51-56
F: Example of triple integral over the region surrounded by the
paraboloid x=y^2+z^2 and plane x=4, when viewed as region of type 1,2 and 3.
Recap: Polar coordinates and integration over polar regions (Section 15.3).
New Material: Cylindrical coordinates (Section 15.7).
Recommended practice problems:
Section 15.6 Exercises 9-18, 19-22, 39-42;
Section 15.3 Exercises 1-4, 5-6, 7-14;
Section 15.7 Exercises 1-2,3-4,5-6,7-8,9-10,11-12
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Last Modified: January 29, 2018