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).

Evaluate $\displaystyle \iint _ { D } x ^ { 2 } d x d y$ ,where D is determined by the two conditions $0 \leq x \leq y$ and $x ^ { 2 } + y ^ { 2 } \leq 1$.

Evaluate the line integral, where C is the given curve.

$$\int_C x e^{y z} d s .$$

$C$ is the line segment from $(0, 0, 0)$ to $(1, 2, 3)$.

Show that the given curve $\mathbf{c}(t)$ is a flow line of the given velocity vector field $\mathbf{F}(x, y, z)$.

$$\mathbf{c}(t) = (\frac{1}{t^3} , e^t ,\frac{1}{t}); \mathbf{F}(x, y, z) = (−3z^4, y, −z^2)$$

Calculate the flow line x(t) of the given vector field F that passes through the indicated point at the specified value of t. $\mathbf{F}(x, y, z)=2 \mathbf{i}-3 y \mathbf{j}+z^{3} \mathbf{k} ; \ \mathbf{x}(0)=(3,5,7)$

Evaluate the line integral, where C is the given curve. integral through C ydx+zdy+xdz, C: x=t^1/2, y=t, z=t^2, 1<=t<=4

If parametric equations of the flow lines are x - x(t), y=y(t) what differential equations do these functions satisfy? Deduce that dy/dx=x.

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).

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 F by drawing a diagram like Fig ure 5 or Figure 9. F(x,y)=yi+xj/(x^2+y^2)^1/2

Find the gradient vector field ∇ f of f and sketch it.

$$f(x, y) = x^2 - y$$

Find the gradient vector field ∇ of f and sketch it.

$$f(x, y) = √x^2+y^2$$

Find the gradient vector field of f. f(x,y,z) =(x^2+y^2+z^2)^1/2

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.