Additional exercises 3

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Exercise

Sketch the vector field $F(x,y)=(-y,x)$.

Exercise

Let $C$ denote the curve $x^2+y^2=4$. Find a clockwise parameterization of the curve $C$.

Exercise

Let $C$ denote the line segment from the point $\mathbf{a}=(1,2,3)$ to the point $\mathbf{b}=(5,4,3)$. Find a parameterization of $C$, starting from $\mathbf{a}$ and finishing at $\mathbf{b}$.

Exercise

Let $C$ denote the curve $x^2+y^2=4$. Find an anticlockwise parameterization of the curve $C$.

Exercise

Let $C$ denote the portion of $y=x^2+2$ from $(-1,3)$ to $(2,6)$. Find a parameterization of the curve $C$.

Exercise

Consider the vector field $F(x,y)=(\sqrt{y},x^3+y)$ and the path $\mathit{\alpha }(t)=(t^2,t^3)$ for $t\in (0,1)$. Evaluate $\int F\cdot d\mathit{\alpha }$.

Exercise

Consider the vector field $F(x,y)=(2-y,x)$ and path $\mathit{\alpha }(t)=(t-\sin t,1-\cos t)$, $t\in [0,2\pi ]$. Compute the line integral $\int F\cdot d\mathit{\alpha }$.

Exercise

The vector field $\mathbf{F}(x,y)=(3x^{2}y,x^3)$ is conservative on ${\mathbf{R}}^2$. Find $\phi$ such that $\mathbf{F}=\mathrm{\nabla }\phi$.

Exercise

Consider the vector field

$$\mathbf{f}(x,y)=\left(\begin{array}{c}-y{(x^2+y^2)}^{-1}\\ x{(x^2+y^2)}^{-1}\end{array}\right)$$

defined on $S={\mathbb{R}}^2\setminus (0,0)$. Let $\mathit{\alpha }(t)$ denote the path which traverses clockwise the circle of radius $r>0$ centred at the origin. Evaluate the line integral $\int \mathbf{f}\cdot d\mathit{\alpha }$.

Exercise

Evaluate $\int (x^2-2y)d\mathit{\alpha }$ where $d\mathit{\alpha }$ is the path defined as $\mathit{\alpha }(t)=(4t^4,t^4)$ for $t\in [-1,0]$.

Exercise

Determine if the vector-field $\mathbf{G}(x,y)=(2y^2,x+2)$ is conservative on ${\mathbb{R}}^2$.

Exercise

Evaluate $\int \mathrm{\nabla }f\cdot d\mathit{\alpha }$ where $f(x,y)=y{e}^{x^2-1}+4xy$ and the path is $\mathit{\alpha }(t)=(1-t,2t^2-2t)$ for $0\le t\le 2$.