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Vector Calculus Exam Questions and Correct Answers Latest Update 2024 (Graded A+)
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Vector Calculus Exam Questions and Correct Answers Latest Update 2024 (Graded A+) Vector Fields - Answers - Let D be set in R^2 (a plane region): vector field on R^2 is a function F that assigns to each point (x,y) in D a 2D vector F(x,y) - F(x,y) = P(x,y)i + Q(x,y)j - two different types of vec...

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  • November 14, 2024
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Vector Calculus Exam Questions and Correct Answers Latest Update 2024 (Graded A+)

Vector Fields - Answers - Let D be set in R^2 (a plane region): vector field on R^2 is a function F that
assigns to each point (x,y) in D a 2D vector F(x,y)

- F(x,y) = P(x,y)i + Q(x,y)j

- two different types of vector fields: source and rotational

- P and Q: scalar fields

-With 3D fields: add one more component R(x,y,z)

Gradient Fields - Answers - delf: actually a vector field called a gradient vector field

-vector field F called conservative vector field it is the gradient of some scalar function (if there exists a
function f such that F = delf)

-f: called potential function of F

Line Integrals - Answers - integrating over a curve C instead of integrating over an interval [a,b]

- if integral C f(x,y) ds = integral from a to b f(x(t),y(t)) ((dx/dt)^2 + (dy/dt)^2)^1/2 dt

-value of the line integral does not depend on the parametrization of the curve provided that curve is
transversed exactly once as t increases from a to b

- s(t): length of C between r(a) and r(t)

-for piecewise-smooth curves: can add the separate integrals up

-physical interpretation: if f(x,y) greater than or equal to 0 then line integral of f(x,y) ds is the area of one
side of the fence of curtain

- line integral of f(x,y) dx = integral from a to b of f(x(t),y(t)) x'(t) dt

- line integral of f(x,y) dy = integral rom a to b of f(x(t),y(t)) y'(t) dt

- vector representation of line segment that starts at r0 and ends at r1 is given r(t) = (1-t)r0 + t r1 and t is
greater than or equal to 0 which is less than or equal to t

-let F be a continuous vector field defined on smooth curve C given by vector function r(t) a is less than
or equal to t which is less than to b: line integral of F along C is line integral of F dot dr = integral from a
to b of F(r(t)) dot r'(t) dt = integral of F dot T ds

- integral -C of F dot dr = - integral C F dot dr because unit tangent vector T is replaced by its negative
when C is replaced by -C

, Fundamental Theorem of Line Integrals - Answers - let C be smooth curve given by vector function r(t)
with t greater than or equal to a which is less than or equal to b

- Let f be differentiable function of two or three variables whose gradient vector del f is continuous on C

- integral C of del f dot dr = f(r(b)) - f(r(a))

- line integral of conservative vector field depends only on initial and terminal point of a curve

- if F is continuous vector field with domain D: line integral integral C of F dot dr is independent of path if
integral C1 of F dot dr = integral C2 of F dot dr

- implication: line integrals of conservative vector fields are independent of path

- curve is called closed if terminal point coincides with initial point (r(b) = r(a))

-conversely: if it is true that integral C F dot dr = 0 whenever C is closed path in D, then demonstrate
independence of path by proving that integral C1 F dot dr = integral C2 F dot dr

- integral C F dot dr is independent of path in D if and only if integral C F dot dr = 0 for every closed path
C in D

- physical interpretation: work done by conservative force field as it moves an object around a closed
path is 0

- following theorem: only vector fields that are independent of path are conservative

- Stated and proved for plane curves

- Furthermore assume that D is connected: means that any two points in D can be joined by path that
lies in D

- Suppose F is a vector field that is continuous on open connected region D: if integral C F dot dr is
independent of path in D, then F is conservative vector field on D

- in other words: there exists a function f such that delf = F

- if F(x,y) = P(x,y) i + Q(x,y) j is conservative vector field where P and Q have continuous 1st-order partial
derivatives on domain D then throughout D we have dP/dy = dQ/dx

-converse: true only for special type of region

- first need concept of simple

Green's Theorem - Answers - gives relationship between line integral around simple closed curve C and
double integral over plane region D bounded by C

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