Thomas’ Calculus Early Transcendentals 12ed

Study Guide for Chap 16

1. Section 16.1: skip the definitions of first moments, center of mass and moments of inertia. Section 16.2: skip the definitions of work, circulation, flow and flux. All exam problems will be expressed in explicit mathematical symbols.

2. Section 16.1, 16.2:

Study the meanings of

∫_{C}f (x, y, z)ds,

∫_{C}F(x, y, z) · Tds = ∫_{C}F(x, y, z) · dr = ∫_{C}M(x, y, z) dx + N(x, y, z) dy + P(x, y, z) dz ,

∫_{C}F(x, y) · nds = _{C}M(x, y) dy − N(x, y) dx

and how to calculate them using a properly chosen parametrization of C: r(t), t_{0} ≤ t ≤ t_{1}. A few points to think over: Which of them is (are) independent of the orientation of C? Which of them depend(s) on the orientation of C? How do you choose the parametrization r(t) so that the direction of T comply with the orientation of C? How is the outward normal n related to T if the parametrization of C is increasing in the counter-clockwise direction?

3. Section 16.3:

Study and memorize the definitions of ’path independent’, ’conservative’ (p938) and ’potential function’ (p939). Study the proof of Theorem 1:’Fundamental Theorem of Line Integrals’ (p940). Study the definition of ’simply connected’ (bottom of p939 and the pictures in p940). Study the proof of Theorem 2: ’Conservative Fields are Gradient Fields’ (p941). Regarding the two theorems on equivalent characterization of conservative fields, Theorem 3: ’Loop Property’ (p942) and ’Component test’ (p943), which of them requires the domain to be simply connected? If the domain is not simply connected (roughly speaking, it has a hole), which of the implications ’⟸’, ’⟹’ remains valid? which one may probably be false? Is Example 5 a counter example of which case?

For given functions M(x, y, z), N(x, y, z), P(x, y, z) satisfying the component test, how does one find the potential function (if it exists) by way of direct integration (example 3)?

4. Section 16.4:

Study and memorize definitions of divergence and curl. It is easier to start with

divF(x, y, z) = ∇ · F(x, y, z), and curlF(x, y, z) = ∇ × F(x, y, z),

then reduce them to

divF(x, y) and curlF(x, y, 0) · k.

Study and memorize Green’s Theorem both in normal form (Theorem 4, p954) and in tangential form (Theorem 5, p955)

Is Theorem 5 applicable to Example 3 of section 16.3? Which part went wrong?

5. Section 16.5, 16.6:

Study how to calculate _{S} G(x, y, z) dσwhen the surface S is given by the parametrization

r(u, v) = f (u, v)i + g(u, v)j + h(u, v)k, a ≤ u ≤ b, c ≤ v ≤ d.

Study how to calculate _{S} G(x, y, z) dσwhen the surface S is given by a graph

x = f (y, z), (y, z) ∈ R,

or a graph

y = g(x, z), (x, z) ∈ R,

or a graph

z = h(x, y), (x, y) ∈ R.