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 $\mathbf{F}(x, y)=x \mathbf{i}-y \mathbf{j}$ to draw some flow lines. From your sketches, can you guess the equations of the flow lines?

A particle moves in a velocity field V(x, y)=⟨x^2, x+y⟩. If it is at position (2, 1) at time t=3, estimate its location at time t = 3.01

Show that the differential equations and in each case find a one-parameter family of integral curves. $2 x y d x+x^{2} d y=0.$

Let

$$I = \iint _ { \mathcal { S } } \mathbf { F } \cdot d \mathbf { S },$$

where

$$\mathbf { F } ( x , y , z ) = \left\langle \frac { 2 y z } { r ^ { 2 } } , - \frac { x z } { r ^ { 2 } } , - \frac { x y } { r ^ { 2 } } \right\rangle \quad \left( r = \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } \right)$$

and

$$\mathcal { S }$$

is the boundary of a region

$$\mathcal { W }.$$

(a) Check that

$$\mathbf { F }$$

is divergence-free. (b) Show that I = 0 if

$$\mathcal { S }$$

if a sphere centered at the origin. Explain, however, why the Divergence Theorem cannot be used to prove this.

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?

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 $\int_C \mathrm{F} \cdot d \mathrm{r}$. 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

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

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.

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

Use polar coordinates and L’Hôpital’s Rule to find the limit. lim_(x, y)→(0, 0) (x² + y²)ln (x² + y²)

Use polar coordinates to find the limit. [If $(r, \theta)$ are polar coordinates of the point $(x, y)$ with $r \geqslant 0$, note that $r \rightarrow 0^{+}$ as $(x, y) \rightarrow(0,0)$.]

$$\lim _{(x, y) \rightarrow(0,0)} \frac{x^3+y^3}{x^2+y^2}$$

Find all second partial derivatives.

$$z=2 x y^3-3 x^2 y$$

Evaluate the double integral $\iint_{D}x\cos y~dA, D$ is bounded by $y = 0, y = x^2, x = 1$.