91Ó°ÊÓ

Berechne die „Gradientenfelder" \(\operatorname{grad} \varphi\) von folgenden Skalarfeldern: (a) \(\varphi(\underline{x})=\underline{a} \cdot \underline{x}\); (b) \(\varphi(\underline{x})=x_{1}^{2}+3 x_{1} x_{2}-x_{2}^{2} ;\) (c) \(\varphi(\underline{x})=|\underline{x}|^{\alpha}(x \neq 0\), falls \(\alpha \leq 1)\); (d) \(\varphi(\underline{x})=e^{|x|}(\underline{x} \neq \underline{0})\).

Zeige, daß \(\varphi(\underline{x})=x_{1}^{m}+x_{2}^{m}+\ldots+x_{n}^{m} \quad(m \in \mathbb{N})\) die Gleichung \(\underline{x} \cdot \operatorname{grad} \varphi(\underline{x})=m \varphi(\underline{x})\) erfüllt.

$$ \int_{K}\left(x_{1}^{2} x_{2} d x_{1}+\left(x_{1}-x_{2} x_{1}\right) d x_{2}\right) $$ $$ K: x_{1}=4 t, x_{2}=3 t \quad(0 \leq t \leq 1) $$

$$ \int_{K}\left(x_{1}^{2}+x_{2}^{2}\right) d s $$ $$ K: x_{1}=t, x_{2}=t^{2} \quad(-2 \leq t \leq 2) $$

$$ \underline{V}(x, y)=e^{x}\left[\begin{array}{l} \sin y \\ \cos y \end{array}\right] $$

$$ \underline{V}(x, y)=\left[\begin{array}{c} x y^{4}+2 x^{5} \\ 2 x^{2} y^{3}-y^{6} \end{array}\right] $$

$$ \underline{V}(x, y, z)=\left[\begin{array}{c} 6 x^{2} \\ 2 x^{3}-z \\ y \end{array}\right] $$

$$ \underline{V}(x, y)=\left[\begin{array}{c} -y \\ x \end{array}\right] $$

$$ \underline{V}(x, y, z)=\left[\begin{array}{l} x \\ 0 \\ 0 \end{array}\right] $$

$$ \underline{V}(x, y, z)=\left[\begin{array}{l} y \\ 0 \\ 0 \end{array}\right] $$