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Mathematics-Online problems:

Interactive Problem 193: true/false: Statements about Potentials

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Let $ D\subseteq\mathbb{R}^2$. Let $ g:D\longrightarrow\mathbb{R}^2$ be a vector field with $ g(x,y)=\left(\begin{array}{c}g_1(x,y)\\ g_2(x,y)\end{array}\right)$.

Let $ E\subseteq\mathbb{R}^3$. Let $ h:E\longrightarrow\mathbb{R}^3$ be a vector field.

Let $ K$ be a smooth curve with the regular parametrization $ C:[0,\pi]\longrightarrow E$.

Decide wether the following statements are true respectively false.

a)
$ g_1(x,y)+g_2(x,y)y'$ is an exact ODE $ \Longrightarrow$ $ g$ has a potential function.

n/a,     true ,     false

b)
$ h$ has a potential function $ \Longrightarrow$ the line integral $ \displaystyle\int\limits_0^\pi h(C(t))\cdot\dot{C}(t)~dt=0$.

n/a,     true ,     false

c)
$ \operatorname{div}(h)=0$ $ \Longleftrightarrow$ $ h$ has a potential function.

n/a,     true ,     false

d)
$ \operatorname{rot}(h)=(0,0,0)^{\operatorname t}$ $ \Longleftrightarrow$ $ h$ has a potential function.

n/a,     true ,     false


   
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