Chap 2 Conditions of Formulas

Question 1

Additivity of Line Integrals

Answer 1

• C must be a connected curve that can be decomposed as C1 U C2 U … U Cn

Question 2

Continuity of Vector Fields

Answer 2

• Each component is continuous

Question 3

Differentiability of Vector Fields

Answer 3

• Each component is differentiable

Question 4

Line Integrals of Vector Field F (and in differential form)

Answer 4

• F is continuous
• Defined along a SMOOTH curve*
• Parametrized by r(t) (a <= t <= b)

Question 5

Fundamental Theorem of Line Integrals (intc grad(f) dot dr = f(r(b)) - f(r(a)))

Answer 5

• C is a smooth curve*
• C is ENTIRELY contained in the domain of f**
• f is continuously differentiable
• f is SCALAR

Question 6

Potential Functions of vector field F

Answer 6

• simply connected domain
• f is scalar
• Curl(F) = 0 or 0 vector*

Question 7

Green’s Theorem on R (posInt(F dot T)ds = (int(int(curlF dA)) = posIntC(Pdx + Qdy))

Answer 7

• R is open
• R is simply connected
• boundary curve C is piecewise
• C is smooth
• C is SIMPLE*
• C is closed
• C is POSITIVELY ORIENTED*
• F has continuous partial derivatives on R**

Question 8

Regions with Holes (resulting in positively oriented region)

Answer 8

• Outer boundary curve must be positively oriented
• Inner boundary curve must be negatively oriented

Question 9

Divergence Theorem (double pos int(S) of F dot d(b sigma)) = triple int(D) grad dot F dV

Answer 9

• F is a vector field with continuous partial derivatives
• S is piecewise smooth
• S is POSITIVELY ORIENTED
• S IS CLOSED**