Problem 1

Evaluate anticlockwise around the square with vertices , , , and .

Problem 2

For each of the following fields, decide which of the following holds:

• A. The characterization of conservative vector fields tells us is conservative.
• B. The characterization of conservative vector fields tells us is not conservative.
• C. The characterization of conservative vector fields does not tell us whether is conservative or not.

(a).

A. is conservative.

(b).

B. is not conservative.

(c).

A. is conservative.

(d).

A. is conservative.

Problem 3

Let , and define . Evaluate over the closed curve that is an ellipse traversed clockwise, centred at , passing through the points , , and .

Problem 4

Let and be points in . Let and be paths from to . Suppose is a conservative vector field in with . What is ?

Problem 5

Let . For which values of the constants , , is for all closed paths ?

Problem 6

Consider the vector field

(a). Determine the Domain of

(b). Compute . Simplify the Result

(c). Evaluate the line Integral where is the Circle of Radius in the Plane , Centred at and Traversed Anticlockwise if Viewed from the Positive -axis (i.e. Viewed from above)

(d). Is Conservative?

No is not conservative as it is not path independent.

Problem 7

Find the work, , done by the force field in moving an object from to . Does the work done depend on the path used to get from to ?

Not the work done is not dependent on the path as the field is a conservative vector field meaning it is path independent.

Problem 8

Consider the vector field

Evaluate the line integral along the oriented curve obtained by moving from to to and finally to along straight line segments.

Problem 9

Evaluate for:

(a). Along from to

(b). Along the Polygonal Path from to to to

Problem 10

Let be the part of the curve of intersection of and which lies between the points and . Calculate

where

Problem 12

Find the work done by the force field on a particle that moves along the line segment from to .

Problem 13

Let . Let be the path which starts at , ends at , and follows

Find the work done in moving a particle along in the field .

Problem 14

Let and let be the line segment for . Evaluate .

Problem 15

Let be the upper half of the unit circle centred on (i.e. that part of the circle which lies above the -axis), oriented clockwise. Compute the line integral .

Problem 16

Show that the following line integral is independent of path and evaluate the integral:

where is any path from to .

The function is irrotational and defined everywhere on it's domain so it's conservative and path independent.

Problem 17

Evaluate the integral

around the triangle with vertices , , and , oriented clockwise as seen from the point .

Problem 18

Evaluate the line integral , where is the conservative vector field

and is the curve given by the parametrization

Problem 19

(a). For Which Values of the Constants , , and is the Vector Field Conservative?

(b). For Those Values of , , and Found in (a), Calculate , where is the Curve Parametrized by , , ,

Problem 20

Consider the vector field .

(a). Compute the Curl of

(b). Is there a Function such that ? Justify Your Answer

Yes there is as the function is a conservative vector field as it's curl is 0 and it's defined on a fully connected domain.

(c). Compute the Integral along the Curve Parametrized by with

Problem 21

(a). Consider the Vector Field Find the Curl of . is Conservative?

Yes is conservative.

(b). Find the Integral of the Field from (a) where is the Curve with Parametrization where Ranges from to

Problem 22

A physicist studies a vector field . From experiments, it is known that is of the form

where and are some real numbers. From theoretical considerations, it is known that is conservative.

(a). Determine and

(b). Find a Potential such that

(c). Evaluate the line Integral where is the Curve Defined by

(d). Evaluate the line Integral where is the Same Curve as in part (c). (Note: the "4" in the Last Term is not a misprint!)

Problem 23

Let

(a). Find All Values of and for Which the Vector Field is Conservative

(b). If and Have the Values Found in (a), Find a Potential Function for

(c). Let Be the Curve with Parametrization from to . Evaluate $$\int_C \big(y^2e^{3z} + xy^3\big),dx + \big(2xye^{3z} + 3x^2y^2\big),dy + 3xy^2e^{3z},dz.$$

Problem 24

(a). For Which value(s) of the Constants and is the Vector Field $$\vec{F} = \big(2x\sin(\pi y) - e^z\big)\hat{i} + \big(ax^2\cos(\pi y) - 3e^z\big)\hat{j} - \big(x + by\big)e^z\hat{k}$$

conservative?

(b). Let Be a Conservative Field from part (a). Find All Functions for Which

(c). Let Be a Conservative Field from part (a). Evaluate where is the Intersection of and from to

(d). Evaluate where $$\vec{G} = \big(2x\sin(\pi y) - e^z\big),\hat{i} + \big(\pi x^2\cos(\pi y) - 3e^z\big),\hat{j} - xe^z,\hat{k}$$

and is the intersection of and from to

Problem 25

Consider the vector field

(a). Find a Real-valued Function such that

(b). Evaluate the line Integral where is the Arc of the Curve , , Traversed from to

Problem 26

Consider the vector field .

(a). Compute

(b). If is Any Path from to and , Show that

Problem 27

Let be the parametrised curve given by

and let .

(a). Compute and Simplify

(b). Compute the Work Integral

Problem 28

(a). Show that the Planar Vector Field $$\vec{F}(x, y) = \big(2xy\cos(x^2),; \sin(x^2) - \sin(y)\big)$$

is conservative

(b). Find a Potential Function for

(c). For the Vector Field from Above, Compute , where is the part of the Graph from to

Problem 29

Consider the following force field, in which , , , are constants:

(a). Find All Values of , , , such that for All Piecewise Smooth Closed Curves in

(b). For Every Possible Choice of , , , in (a), Find the Work Done by in Moving a Particle from the Bottom to the top of the Sphere . (The Direction of Defines "up".)

Problem 30

A particle of mass has position and velocity at time . The particle moves under a force

where denotes time.

(a). Find the Position of the Particle as a Function of

(b). Find the Position of the Particle when it Crosses the Plane for the First time at

(c). Determine the Work Done by in Moving the Particle from to

Problem 31

Let be the curve from to along the intersection of the surfaces and .

(a). Find if is Arc Length along and

(b). Find if

Problem 32

The curve is the helix that winds around the cylinder (anticlockwise, as viewed from the positive -axis, looking down on the -plane). It starts at the point , winds around the cylinder once, and ends at the point . Compute the line integral of the vector field

along .

Problem 33

Evaluate the line integral , Where is the Arc of the Parabola from to

Problem 34

A particle of mass has position and velocity at time . The particle moves under a force , where denotes time.

(a). Find the Position of the Particle as a Function of

(b). Find the Position of the Particle when it Crosses the Plane for the First time after time

(c). Determine the Work Done by in Moving the Particle from to

Problem 35

(a). Consider the Vector Field in . Compute the line Integral where is the line Segment from to

(b). Find an Oriented Path from to such that where is the Vector Field from (a)

The path giving us is the path

Problem 36

Let be a vector field on . Find an oriented curve from to such that .

The path giving us is the path .

Problem 37

Let and suppose that is a function defined everywhere with everywhere continuous partial derivatives. Show that for any curve whose endpoints and lie on the -axis,

Because is defined everywhere and is irrotational must be a conservative vector field and is thus path independent so any line integral of over the path will be equal to the straight path from to because and are both on the -axis we can ignore the term as it will never contribute leaving us just integrating 1 over the path from to which gives us the distance between the two.

Problem 38

Let be the surface and let . Consider the points and on the surface .

Find a value of the constant so that for any two curves and on the surface from to .

Consider the vector function which we know is conservative with a potential function .

Hence if is conservative over the surface then than any line integrals over a curve on must be path independent.

Problem 39

Consider the vector field

where and are real-valued constants.

(a). Compute

(b). Determine for Which Values of and the Vector Field is Conservative

(c). For the Values of and Obtained in part (b), Find a Potential Function such that

(d). Evaluate where is the Arc of the Curve Starting at and Ending at

Problem 40

The vector field is conservative on .

(a). Find the Values of the Constants and

(b). Find a Potential such that on

(c). If is the Curve , from to , Evaluate

(d). Evaluate where is the Curve of part (c)

(e). Let Be the Closed Triangular Path with Vertices , , and , Oriented Anticlockwise as Seen from the point . Evaluate

Problem 41

A particle of mass is acted on by a force . At , the particle has velocity zero and is located at the point .

(a). Find the Velocity Vector for

(b). Find the Position Vector for

(c). Find , the Curvature of the Path Traversed by the Particle for

(d). Find the Work Done by the Force on the Particle from to

Problem 42

The position of an airplane at time is given by

from take-off at to landing at .

(a). What is the Total Distance the Plane Travels on This Flight?

(b). Find the Radius of Curvature at the Apex of the Flight, Which Occurs at

(c). Two External Forces Are Applied to the Plane during the Flight: the Force of Gravity , where is the Mass of the Plane and is a Constant; and a Friction Force , where is the Velocity of the Plane. Find the Work Done by Each of These Forces during the Flight

Gravity is a conservative force so because we start and end at the same height there is no work done by the field on the plane.

(d). One Half-hour Later, a Bird Follows the Exact Same Flight Path as the Plane, Travelling at a Constant Speed . One Can Show that at the Apex of the Path, i.e. when the Bird is at , the Principal Unit Normal to the Path Points in the Direction. Find the Bird's (vector) Acceleration at that Moment