Problem 1
Let be the two-dimensional vector field shown below.
(a). Assuming that the Vector Field in the Picture is a Force Field, the Work Done by the Vector Field on a Particle Moving from point to along the given Path is
- (A) Positive
- (B) Negative
- (C) Zero
- (D) Not enough information to determine.
(b). Which Statement is the Most True about the line Integral
- (A)
- (B)
- (C)
- (D) Not enough information to determine.
(c). At the point (in the picture) is
- (A) Positive
- (B) Negative
- (C) Zero
- (D) Not enough information to determine.
(d). At the point is
- (A) Positive
- (B) Negative
- (C) Zero
- (D) Not enough information to determine.
(e). Assuming that , Which of the following Statements is Correct about at the point ?
- (A) at .
- (B) at .
- (C) at .
- (D) The sign of at cannot be determined by the given information.
Problem 2
Does have to be perpendicular to ?
No it does not to be take the counter example above they are not parallel for every point not at the origin.
Problem 3
Verify the vector identities:
(a).
(b).
(c).
Problem 4
Evaluate and for each of the following vector fields.
(a).
(b).
(c). (the Polar Basis Vector in 2D)
(d). (the Polar Basis Vector in 2D)
Problem 5
(a). Compute and Simplify for and . Express Your Answer in terms of
(b). Compute
Problem 6
In the following, we use the notation , , and is some number
(a). Find the Value of for Which
(b). Find the Value of for Which
(c). Find the Value of for Which
Problem 7
Let and . Let be the constant vector . Compute and simplify the following quantities. Answers must be expressed in terms of , , and . There should be no 's, 's, or 's in your answers.
(a).
(b).
(c).
(d).
Problem 8
Let and .
(a). Compute where
(b). Compute where
(c). Compute where
Problem 9
Find, if possible, a vector field that has component and that is a vector potential for:
(a).
There is no vector potential for
(b).
Problem 10
Let
(a). Determine the Domain of
(b). Determine the Curl of . Simplify if Possible
(c). Determine the Divergence of . Simplify if Possible
(d). Is Conservative? Give a Reason for Your Answer
No is not conservative as it's curl is both non-zero as well as it is not defined on the simply connected region of it's domain.
Problem 11
A physicist studies a vector field in her lab. She knows from theoretical considerations that must be of the form , for some smooth vector field . Experiments also show that must be of the form
where and are constants.
(b). Further Experiments Show that . Find the Unknown Function
Problem 12
A rigid body rotates at an angular velocity of rad/sec about an axis that passes through the origin and has direction . When you are standing at the head of looking towards the origin, the rotation is anticlockwise. Set .
(a). Show that the Velocity of the point on the Body is
(b). Evaluate and , Treating as a Constant
(c). Find the Speed of the Students in a Classroom Located at Latitude N due to the Rotation of the Earth. Ignore the Motion of the Earth about the Sun, the Sun in the Galaxy, and so On. the Radius of the Earth is Km
Problem 13
Suppose that the vector field obeys in all of . Let
be a parametrization of the line segment from the origin to . Define
Show that throughout .