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
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
