The flow lines (or streamlines) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus the vectors in a vector field are tangent to the flow lines. Use a sketch of the vector field $\text{F}(x, y) = x\text{i} - y\text{j}$ to draw some flow lines. From your sketches, can you guess the equations of the flow lines?

Sketch the vector field $\mathbf{F}(x, y)=\mathbf{i}+x \mathbf{j}$ and then sketch some flow lines. What shape do these flow lines appear to have?

Find curl v for v given with respect to right-handed Cartesian coordinates. Show the details of your work. v=xyz[x, y, z]

The pressure difference between an oil pipe and a water pipe is measured by a double-fluid manometer, as shown in the figure. For the given fluid heights and specific gravities, find the pressure difference $\Delta P=P_B-P_A$.

A $30^{\circ}$ reducing elbow is shown. The fluid is water. Evaluate the components of force that must be provided by the adjacent pipes to keep the elbow from moving.

Use Stokes' Theorem to evaluate $\int_C \mathrm{F} \cdot d \mathrm{r}$. In each case $C$ is oriented counterclockwise as viewed from above. Let

$$\mathrm{F}(x, y, z)=x y \mathrm{i}+2 z \mathrm{j}+3 y \mathrm{k}$$

where $C$ is the curve of intersection of the plane $x+z=5$ and the cylinder $x^2+y^2=9$.

Use Stokes' Theorem to evaluate $\int_C \mathbf{F} \cdot d \mathbf{r}$, where $\mathbf{F}(x, y, z)=x^2 y \mathbf{i}+\frac{1}{3} x^3 \mathbf{j}+x y \mathbf{k}$ and $C$ is the curve of intersection of the hyperbolic paraboloid $z=y^2-x^2$ and the cylinder $x^2+y^2=1$ oriented counterclockwise as viewed from above.

Use Stokes' Theorem to evaluate $\int_{C} \mathbf{F} \cdot d \mathbf{r},$ where

$$\mathbf{F}(x, y, z)=x^{2} z\ \mathbf{i}+x y^{2}\ \mathbf{j}+z^{2}\ \mathbf{k}$$

and C is the curve of intersection of the plane x + y + z = 1 and the cylinder $x^{2}+y^{2}=9$ oriented counterclockwise as viewed from above.

Use Stokes’ Theorem to evaluate ∫cF · dr, where

$$F(x,y,z) = x^2yi + 1/3x^3j+xyk$$

and C is the curve of intersection of the hyperbolic paraboloid

$$z=y^2-x^2$$

and the cylinder

$$x^2+y^2=1$$

oriented counterclockwise as viewed from above.

The y component of velocity in a two-dimensional, incompressible flow field is given by v = −Axy, where υ is in m/s, x and y are in meters, and A is a dimensional constant. There is no velocity component or variation in the z direction. Determine the dimensions of the constant, A. Find the simplest x component of velocity in this flow field. Calculate the acceleration of a fluid particle at point (x, y) = (1, 2).

Find the directional derivative of $f ( x , y ) = \frac { x } { x + y }$ at $P ( 1,0 )$ in the direction of $Q ( - 1 , - 1 )$

Find the volume of the solid in the first octant bounded by the cylinder $z=9-y^2$ and the plane $x=2$.

Consider the rotation field $F=\frac { \langle - y , x \rangle } { x ^ { 2 } + y ^ { 2 } }$.

Use a line integral to verify that the circulation on the unit circle of the vector field is $2 \pi$.

For $\vec{r} = x\vec{i} + y\vec{j} + z\vec{k}$, an arbitrary function $f(x, y, z)$ and an arbitrary vector field, $\vec{F} (x, y, z)$, which of the following is a vector field, and which is a constant vector field?

grad $f$