6.1 Vector fields  (Page 8/15)

The domain of vector field $\text{F}=\text{F}\left(x,y\right)$ is a set of points $\left(x,y\right)$ in a plane, and the range of F is a set of what in the plane?

Vector field $\text{F}=⟨3x^2,1⟩$ is a gradient field for both ${\varphi }_1\left(x,y\right)=x^3+y$ and ${\varphi }_2\left(x,y\right)=y+x^3+100.$

Vector field $\text{F}=\frac{⟨y,x⟩}{\sqrt{x^2+y^2}}$ is constant in direction and magnitude on a unit circle.

Vector field $\text{F}=\frac{⟨y,x⟩}{\sqrt{x^2+y^2}}$ is neither a radial field nor a rotation.

[T] $\text{F}(x,y)=x\text{i}+y\text{j}$

[T] $\text{F}(x,y)=\text{−}y\text{i}+x\text{j}$

[T] $\text{F}(x,y)=x\text{i}-y\text{j}$

[T] $\text{F}(x,y)=\text{i}+\text{j}$

[T] $\text{F}(x,y)=2x\text{i}+3y\text{j}$

[T] $\text{F}(x,y)=3\text{i}+x\text{j}$

[T] $\text{F}(x,y)=y\text{i}+\text{sin}x\text{j}$

[T] $\text{F}(x,y,z)=x\text{i}+y\text{j}+z\text{k}$

[T] $\text{F}(x,y,z)=2x\text{i}-2y\text{j}-2z\text{k}$

[T] $\text{F}(x,y,z)=\frac{y}{z}\text{i}-\frac{x}{z}\text{j}$

$f(x,y)=x\text{sin}y+\text{cos}y$

$f(x,y,z)=z{e}^{\text{−}xy}$

$f(x,y,z)=x^{2}y+xy+y^{2}z$

$f(x,y)=x^2\text{sin}(5y)$

$f(x,y)=\text{ln}\left(1+x^2+2y^2\right)$

$f(x,y,z)=x\text{cos}\left(\frac{y}{z}\right)$

What is vector field $\text{F}\left(x,y\right)$ with a value at $\left(x,y\right)$ that is of unit length and points toward $\left(1,0\right)?$

All vectors are parallel to the x -axis and all vectors on a vertical line have the same magnitude.

All vectors point toward the origin and have constant length.

All vectors are of unit length and are perpendicular to the position vector at that point.

Give a formula $\text{F}(x,y)=M(x,y)\text{i}+N(x,y)\text{j}$ for the vector field in a plane that has the properties that $\text{F}=0$ at $\left(0,0\right)$ and that at any other point $\left(a,b\right),$ F is tangent to circle $x^2+y^2=a^2+b^2$ and points in the clockwise direction with magnitude $\left|\text{F}\right|=\sqrt{a^2+b^2}.$

Is vector field $\text{F}(x,y)=\left(P(x,y),Q(x,y)\right)=\left(\text{sin}x+y\right)\text{i}+\left(\text{cos}y+x\right)\text{j}$ a gradient field?

Find a formula for vector field $\text{F}(x,y)=\text{M}(x,y)\text{i}+N(x,y)\text{j}$ given the fact that for all points $(x,y),$ F points toward the origin and $\left|\text{F}\right|=\frac{10}{x^2+y^2}.$

Find the components of the electric field in the x - and y -directions, where $\text{E}\left(x,y\right)=\text{−}\text{∇}V\left(x,y\right).$

Show that the electric field at a point in the xy -plane is directed outward from the origin and has magnitude $\text{E}|=\frac{c}{r},$ where $r=\sqrt{x^2=y^2}.$

$\text{c}\left(t\right)=\left(e^{2t},\text{ln}\left|t\right|,\frac{1}{t}\right),t\ne 0;\text{F}\left(x,y,z\right)=⟨2x,z,\text{−}{z}^2⟩$

$\text{c}\left(t\right)=\left(\text{sin}t,\text{cos}t,e^t\right);\text{F}\left(x,y,z\right)=⟨y,\text{−}x,z⟩$