Math 3202 represents the culmination of the introductory calculus sequence
which began in Math 1000. Not only does it introduce further topics in
differentiation and integration, but it also incorporates the concept of
the vector as introduced in Math 2050. By considering vector functions --
functions with both magnitude and direction -- we are better able to
characterise different kinds of curves and surfaces, leading to the
critical notions of line integrals and surface integrals. To assist in our
exploration of these concepts, we introduce key results in vector calculus
such as Green's Theorem, Stokes' Theorem, and the Divergence Theorem, all
of which have enormous implications for the study of subjects as diverse
as electromagnetism, fluid dynamics, and black hole astrophysics.
If you have any questions about the course, I can always be contacted at
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- This course has now concluded. Best of luck with your future
studies.
- Assignment 1
(due Friday, May 24th)
- Assignment 2
(due Friday, May 31st)
- Assignment 3
(due Friday, June 7th)
- Assignment 4
(due Friday, June 21st)
- Assignment 5
(due Friday, July 5th)
- Assignment 6
(due Friday, July 12th)
- Assignment 7
(due Friday, July 26th)
- Assignment 8
(due Friday, August 2nd)
- Worksheet
(not to be submitted)
- Test 1
(written Monday, June 17th)
- Test 2
(written Monday, July 22nd)
I do not make lecture notes available outside of class; if you miss a
lecture for any reason, I strongly encourage you to borrow the
corresponding notes from a classmate. However, as a guide, here is a brief
summary of what was covered in each lecture.
- May 13th: Two- and three-dimensional vector functions, notations for
vector functions, the domain of a vector function (Section
1.1)
- May 15th: Vector operations, graphs of vector functions and space
curves, parametrisations (Section 1.1)
- May 17th: Limits and continuity for vector functions, derivatives of
vector functions and their properties, the vector derivative as a tangent
vector (Section 1.2)
- May 20th: Unit tangent vectors, smooth curves, integrals of vector
functions (Section 1.2); calculating the arclength of a curve
(Section 1.3)
- May 22nd: Arclength functions, reparametrisations in terms of
arclength, arclength parametrisations and unit tangent vectors
(Section 1.3)
- May 24th: Line integrals, line integrals over piecewise smooth curves
(Section 1.4)
- May 27th: Line integrals over space curves (Section 1.4);
curvature (Section 1.5)
- May 29th: The orthogonality of the unit tangent vector and its
derivative, the unit normal and binormal vectors, the osculating plane and
osculating circle (Section 1.5)
- May 31st: Curvature of a plane curve defined by a function (Section
1.5); position, velocity and acceleration vector functions (Section
1.6)
- June 3rd: Tangential and normal components of acceleration (Section
1.6)
- June 5th: The vector equation of the plane, quadric surfaces, spheres,
(circular) cylinders, ellipsoids (Section 2.1)
- June 7th: Hyperbolic paraboloids, elliptical paraboloids, hyperboloids
(including cones), parametric equations of a surface, the tangent plane to
a surface (Section 2.1)
- June 12th: Finding the equation of a tangent place (Section
2.1); the directional derivative, the gradient vector (Section
2.2)
- June 14th: Finding the direction of greatest increase, the
orthogonality of the gradient vector to level curves, gradients in three
variables, level surfaces (Section 2.2)
- June 19th: Finding tangent planes by treating surfaces as level
surfaces, the normal line (Section 2.2)
- June 21st: Calculating the surface area of a surface (Section
2.3)
- June 28th: Calculating the surface area of a parametrised surface,
surface integrals (Section 2.3); triple integrals (Section
2.4)
- July 3rd: Computing volume using a triple integral (Section
2.4); cylindrical coordinates, simple graphs in cylindrical
coordinates (Section 2.5)
- July 5th: Triple integrals in cylindrical coordinates (Section
2.5); spherical coordinates, simple graphs in spherical coordinates
(Section 2.6)
- July 8th: Triple integrals in spherical coordinates (Section
2.6); vector fields, vector line integrals (Section 3.1)
- July 10th: Computing vector line integrals, vector surface
integrals (Section 3.1)
- July 12th: Computing vector surface integrals, vector surface
integrals over a surface defined by a function (Section 3.1);
conservative vector fields and potential functions, the
Fundamental Theorem of Line Integrals (Section 3.2)
- July 15th: Independence of path, closed curves and circulation,
equivalence of conservative vector fields and path-independence
(Section 3.2)
- July 17th: Equivalence of path-independence and zero circulation,
conditions for conservative vector fields, determing the potential
function of a two-dimensional vector field (Section 3.2)
- July 19th: Determining the potential function of a three-dimensional
vector field (Section 3.2); Green's Theorem (Section
3.3)
- July 24th: Using Green's Theorem to rewrite a line integral as a
double integral (Section 3.3)
- July 26th: Using Green's Theorem to compute area, Green's Theorem
applied to regions that are not simply connected (Section 3.3); the
curl of a vector field, Stokes' Theorem in two dimensions (Section
3.4)
- July 29th: The divergence of a vector field, the Divergence Theorem in
two dimensions (Section 3.4); Stokes' Theorem in three dimensions,
orientation of a boundary curve relative to a surface (Section
3.5)
- July 31st: Applying Stokes' Theorem (Section 3.5); the
Divergence Theorem in three dimensions (Section 3.6)
- August 2nd: Applying the Divergence Theorem (Section 3.6);
Lagrange multipliers in two variables (Section 4.1)
- August 9th: Lagrange multipliers in three variables (Section
4.1)
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