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. If parametric equations of a flow line are $x = x(t), y = y(t)$ explain why these functions satisfy the differential equations $dx/dt = x$ and $dy/dt = -y$. Then solve the differential equations to find an equation of the flow line that passes through the point (1, 1).

In this exercise, compute the indefinite integral of the following function.

r(t)=$2^t i+\dfrac{1}{1+2 t} j+\ln t k$

Compute the indefinite integral of the following functions. r(t)=$2 ^ { t } \mathbf { i } + \frac { 1 } { 1 + 2 t } \mathbf { j } + \ln t \mathbf { k }$

$\vec{F} = (e^z + x)\vec{i} + (2y + \sin x)\vec{j} +\Big (e^{x^2}− z \Big)\vec{k}$ and $S$ is the sphere of radius $1$ centered at $(2, 1, 0)$.

Find the center of mass of the hemisphere $x^2+y^2+z^2=a^2, z \geqslant 0$, if it has constant density.

Use Green’s first identity to show that if $f$ is harmonic on $D,$ and if $f(x, y) = 0$ on the boundary curve $C,$ then $\iint_D\lvert \triangledown f\rvert^2~dA = 0$. (Assume the same hypotheses as in the previous exercise.)

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

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?

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 F(x, y) = xi - yj to draw some flow lines. From your sketches, can you guess the equations of the flow lines?

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.