This is the strict minimum that a student finishing Math 226 should
know.
The suggested problems are straightforward applications of the material. Chapter 13 Section 13.1
What it covers: This section defines the cartesian coordinate
system in the three dimensional space.
What you need to know: The distance formula (page 831), and the
equation for a sphere (page 832)
What you need to do: Problems 7, 11, 17. Section 13.2
What it covers: Definition of vectors, how to add
subtract, multiply with a scalar, components.
What you need to know: components, how to add
vectors, multiply a vector with a scalar, components, magnitude,
standard basis.
What you need to do: Problems 13, 15, 17 Section 13.3
What it covers: The dot product
What you need to know: Formulas for the dot product (in terms of
components and in terms of magnitudes and angles), Direction angle,
unit vector, how to find the angle between two vectors. Projections may
be omitted.
What you need to do: Problems 3, 5, 7, 17, 29. Section 13.4
What it covers: The cross product
What you need to know: Formula of the cross product in terms of
the determinant. The cross product of two vectors is a vector
orthogonal to both vectors. The length of the cross product formula
in terms of magnitudes and the sine of the angle (page 852). The area
of
a parallelogram given by two vectors is the length of the cross product
of the two vectors (page 853). The volume of the parallelepiped formula
(11) page 855
What you need to do: Problems 1, 3, 13, 25, 29, Section 13.5
What it covers: Equations of lines and planes
What you need to know: Formula (1) page 858 giving the vector
equation of the line, Formula (2) giving the parametric equation of the
line, Formula (4) page 860 for the line segment. Vector equation
of the plane (Formula (5)), as well as the Cartesian equation of
the plane (Formulas (7) and (8)) on page 862. You need to be able to
write an equation of the plane if you know 3 points on it, a
point and two vectors on it, or a point in it and a normal vector to
it. Direction numbers. Formula for the distance of a point to
plane (Formula (9)) page 864.
What you need to do: Problems 3, 5, 17, 23,
25, 27, 29, 37, 65, 67 Section 13.7
What it covers: Cylindrical and spherical coordinates.
What you need to know: Cylindrical and spherical coordinates
Formulas (1), (2), (3) and (4).
What you need to do: Understand the formulas. Chapter 14
Section 14.1
What it covers: Vector functions and space curves
What you need to know: What is a vector function and how to
describe a space curve.
What you need to do: Nothing Section 14.2
What it covers: Derivatives and integrals of vector functions
What you need to know: How to differentiate and integrate vector
functions (component by component). How to find a tangent vector to
curve (see Example 1, page 894)
What you need to do: Problems; 9, 11, 17, 19, 23, 33, 35 Section 14.3
What it covers: Arc length (but not Curvature)
What you need to know: Arc length Formulas (1)-(3), as well as
the arc length function (Formulas (6) and (7)) page 899.
What you need to do: Problems: 1, 3, 9. Section 14.4
What it covers: Velocity and acceleration
What you need to know: Velocity is the first derivative and
acceleration is the second derivative.
What you need to do: Problems Nothing Chapter 15
Section 15.1
What it covers: Functions of several variables
What you need to know: Domain and level sets
What you need to do: Problems: 11, 13, Section 15.2
What it covers: Limits and Continuity
What you need to know: How to compute limits, when is a function
continuous.
What you need to do: Problems: 7, 9, 11, 13, 15, 27, 29. Section 15.3
What it covers: Partial Derivatives
What you need to know: Basic definition of partial derivatives
(Formulas (2) -(4) page 947). Interpretation of partial derivatives as
rates of change, higher order partial derivatives, Clairaut's Theorem
and what is a solution to a partial differential equation (page 953).
What you need to do: Problems: 13, 17, 25, 35, 47, 67. Section 15.4
What it covers: Tangent planes and linear approximations
What you need to know: Equation of a tangent plane to a surface
z=f(x,y) (Formula (2) page 959). The linearization of a function
(Formula (4) ) page 961. Differential (Formula (10)).
What you need to do: Problems: 1, 3, 5. Section 15.5
What it covers: The chain rule
What you need to know: The chain rule case I (Formula (2) page
968, and case II, Formula (3).
What you need to do: Problems: 1, 3, 7, 9, 21, 23. Section 15.6
What it covers: Directional derivatives and the Gradient vector.
What you need to know: Definition of the directional derivative
page 977, Formula (3) which gives you the directional derivative
in term of partial derivatives. Definition of the gradient vector
(Formula (8)), and Formula (9) that gives the directional derivative in
terms of the gradient. Maximizing the directional derivative Theorem
(15), page 982. Tangent planes to level surfaces (Formula (19)).
What you need to do: Problems: 5, 7, 9, 11, 13, 21. Section 15.7
What it covers: Maximum and minimum values
What you need to know: What is a local min/max, Theorem (2) page
989. What is a critical point. The second derivative test. How to find
the absolute maximum and minimum values. Box (9) page 995
What you need to do: Problems: 5, 7, 11, 27, 29 Section 15.8
What it covers: Lagrange multipliers
What you need to know: The method of Lagrange multipliers page
1002
What you need to do: Problems: 3, 5, 9 Chapter 16
Section 16.1
What it covers: Double integral over rectangles
What you need to know: Definition in Box (5) page 1019.
What you need to do: Understand the computation of the volume
under a a surface, i.e the meaning of the double integral. Section 16.2
What it covers: Iterated integrals
What you need to know: Fubini's Theorem. How to compute a double
integral by computing an iterate simple integrals
What you need to do: Problems: 3, 5, 7, 13, 15 Section 16.3
What it covers: Double integrals over general regions
What you need to know: Formulas (3) and (5), Formulas (9)-(11)
What you need to do: Problems: 1, 3, 7, 9, 15, 43, 45 Section 16.4
What it covers: Double integrals in polar coordinates
What you need to know: The change to polar coordinates in the
double integral Formula (Box 2, page 1041) , and Formula (3).
What you need to do: Problems: 9, 11, 15, 29, 31 Section 16.5
What it covers: Application of double integral
What you need to know: Surface density and mass
What you need to do: nothing Section 16.6
What it covers: Surface Area
What you need to know: Formula (2), page 1056
What you need to do: Problems: 7, 9 Section 16.7
What it covers: Triple integrals
What you need to know: Definition of the triple integral Formula
3, page 1059, Fubini's Theorem, Iterated integrals.
What you need to do: Problems: 7, 9, 11. Section 16.8
What it covers: triple integrals in cylindrical and spherical
coordinates.
What you need to know: Formula (2) for the change to cylindrical
coordinates, and Formula (4) for the change to spherical coordinates.
What you need to do: Problems: 7, 9, 13, 17, 19, 21, 33, 35. Section 16.9
What it covers: Change of variables in multiple integrals
What you need to know: Definition of a Jacobian of a
transformation (Formula (7)) page 1079, look at Figures 4 and 5 and the
calculations corresponding to them. Then Formulas (8), (9), and (13)
What you need to do: Problems: 1, 3, 7, 11, 13. Chapter 17
Section 17.1
What it covers: Vector Fields
What you need to know: Definition of a vector field, what is a
gradient field, conservative field
What you need to do: Problems: 1, 3, 15, 17, 21, 25 Section 17.2
What it covers: Line integrals
What you need to know: Definitions of line integrals; one with respect
to arc length Formula 3, one with respect to dx, dy, and dz (Formula
(7)), all in terms of a parametrization of the curve. Line integrals of
vector fields (Box (13) page 1106.
What you need to do: Problems: 3, 9, 11, 19, 21, Section 17.3
What it covers: The fundamental theorem for line integrals
What you need to know: Theorem (2), Independence of path
(Theorems(3),-(6)), What is an open set? connected set? a simply
connected set?
What you need to do: Problems: 3, 5, 9, 13, 15, 19, 29, 31 Section 17.4
What it covers: Green's Theorem
What you need to know: Green's Theorem (page 1119), and how to use it.
What you need to do: Problems: 1, 3, 5, 7, 13, Section 17.5
What it covers: Curl and Divergence
What you need to know: You need to know how to compute a curl of a
vector field (Formula (1) or (2)) as well as the divergence of a vector
field
(Formula (9) or (10).
What you need to do: Problems: 1, 3, 5, 7, (good abstract
problems 23-29, 33). Section 17.6
What it covers: Parametric surfaces and their areas
What you need to know: What is a parametric surface, how to find an
equation of a tangent plane to a parametric surface, surface area
(Definition (6)).
What you need to do: Problems: 11, 13, 17, 19, 23, 31, 33, 35, 41. Section 17.7
What it covers: Surface integrals
What you need to know: Definition of surface integrals (formulas (1),
(2)), and especially (3), what is an oriented surface, closed surface,
and a surface integral of a vector field Definition (7) page 1151 and
Formula (9) giving the surface integral in terms of a parametric
representation of the surface. Flux
What you need to do: Problems: 5, 7, 13, 19, 21, 27, Section 17.8.
What it covers: The Stokes Theorem
What you need to know: Stokes Theorem (page 1157), what is circulation
What you need to do: Problems: 1, 3, 5, 7, 9, 13. Section 17.9
What it covers: The divergence Theorem
What you need to know: The divergence Theorem (page 1163)
What you need to do: Problems: 3, 5, 7, 9, 11, 15, 19, 21,
22.