Q: If $$\vec{v}(x, y) is a $$rotational vector field, which of the following must be true for its potential function $$\phi(x, y)$$?

$$\nabla \phi(x, y) is $$not defined everywhere in the domain$

Q: For the vector field $$\vec{v}(x, y) = \nabla \phi(x, y)$$, which of the following is always true?

$$\nabla \times \vec{v}(x, y) = 0$$

Q: If $$\vec{v}(x, y) is a $$velocity field describing incompressible flow, which mathematical condition must be satisfied?

$$\nabla \cdot \vec{v}(x, y) = 0$$

Question 1

Given the velocity field $$\vec{v}(x, y) = U \left(1 + \frac{R^2$}{x^2$ + y^2$}\right) \hat{i}$$ for $$x^2$$ + $$y^2$$ \geq $$R^2$$, which of the following statements is true about the flow at the boundary $$x^2$$ + $$y^2$$ = $$R^2$$?

Accepted Answer

The velocity is tangent to the boundary at every point.

Question 2

For the velocity field $$\vec{v}(x, y) = U \left(1 + \frac{R^2$}{x^2$ + y^2$}\right) \hat{i}$$, what is the value of $$\vec{v} at $$the point $$(0, R)$$?

Accepted Answer

$$2U \hat{i}$$

Question 3

If $$\vec{v}(x, y) is a $$rotational vector field, which of the following must be true for its potential function $$\phi(x, y)$$?

Accepted Answer

$$
abla \phi(x, y) is $$not defined everywhere in the domain$

Question 4

Consider the vector field $$\vec{v}(x, y) = -\sin(x)e$$^{y}$$ \hat{i} + \cos(x)e$$^{y}$$ \hat{j}. $$Which of the following is a potential function for $$\vec{v}$$?

Accepted Answer

$$e^{y}$$\cos(x)$$

Question 5

Given $$\vec{v}(x, y) = U \left(1 + \frac{R^2$}{x^2$ + y^2$}\right) \hat{i}$$, what is $$\frac{\partial v_x}{\partial x} at$$ $$(x, y) = (R, 0)$$?

Accepted Answer

$$-2U \frac{R^2$}{R^3$}$$

Question 6

Which of the following best describes the physical meaning of the streamlines shown in Figure 1 (flow around a cylinder)?

Accepted Answer

They indicate the direction of the velocity vector at every point in the flow.

Question 7

For the vector field $$\vec{v}(x, y) = 
abla \phi(x, y)$$, which of the following is always true?

Accepted Answer

$$
abla 	imes \vec{v}(x, y) = 0$$

Question 8

Suppose $$\vec{v}(x, y) is $$given by $$\vec{v}(x, y) = \frac{\partial \phi}{\partial x} \hat{i} + \frac{\partial \phi}{\partial y} \hat{j}$$, and $$\phi(x, y) = x^2$y. $$What is $$\vec{v}(1, 2)$$?

Accepted Answer

$$4 \hat{i} + 1 \hat{j}$$

Question 9

If $$\vec{v}(x, y) is a $$velocity field describing incompressible flow, which mathematical condition must be satisfied?

Accepted Answer

$$
abla \cdot \vec{v}(x, y) = 0$$

Question 10

Given the potential function $$\phi(x, y) = U r \cos(	heta) \left(1 + \frac{R^2$}{r^2$}ight) in $$polar coordinates, what is the velocity component in the $$	heta$$ direction at $$r = R$$?

Accepted Answer

$$0$$

Question 11

Which of the following statements about the vector field $$\vec{v}(x, y) = U \left(1 + \frac{R^2$}{x^2$ + y^2$}\right) \hat{i} is $$correct?

Accepted Answer

It is irrotational everywhere except possibly at the origin.

Question 12

If $$\vec{v}(x, y) is a $$conservative vector field, which of the following line integrals between two points $$A$$ and $$B is $$always true?

Accepted Answer

The line integral depends only on the endpoints $$A$$ and $$B$$, not the path taken.

Question 13

For the vector field $$\vec{v}(x, y) = -\sin(x)e$$^{y}$$ \hat{i} + \cos(x)e$$^{y}$$ \hat{j}$$, what is $$\frac{\partial v_y}{\partial x}$$?

Accepted Answer

$$-\sin(x)e$$^{y}$$

Question 14

Suppose $$\vec{v}(x, y) is a $$velocity field around a cylinder of radius $$R. $$Which boundary condition must be satisfied at $$r = R$$?

Accepted Answer

The normal component of $$\vec{v} at$$ $$r = R$$ must be zero.

Question 15

Given $$\vec{v}(x, y) = U \left(1 + \frac{R^2$}{x^2$ + y^2$}ight) \hat{i}$$, what happens to $$\vec{v} as$$ $$x^2$$ + $$y^2$$ 	o \infty$$?

Accepted Answer

$$\vec{v} 	o U \hat{i}$$

Question 16

If $$\vec{v}(x, y) = 
abla \phi(x, y)$$ and $$\phi(x, y) = x^2$ + y^2$, what is the curl of $$\vec{v}$$?

Accepted Answer

$$0$$

Question 17

Which of the following is a necessary condition for a vector field $$\vec{v}(x, y) to be $$expressible as the gradient of a scalar potential $$\phi(x, y)$$?

Accepted Answer

$$\frac{\partial v_x}{\partial y} = \frac{\partial v_y}{\partial x}$$

Question 18

In the context of flow around a cylinder, what does the condition $$\vec{v} \cdot \vec{n} = 0 at$$ $$r = R$$ physically represent?

Accepted Answer

No fluid crosses the surface of the cylinder.

Vector Fields and Potential Functions in Fluid Flow: Calculus Applications

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