ベクトル解析2 gradの問題

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ベクトル解析 gradの問題

 

 

問題

① \(\nabla r\)     ② \(\nabla \displaystyle\frac{1}{r}\)     ③ \(\nabla r^n\)

 

※\(r=\sqrt{x^2+y^2+z^2}\)

 

解答

1番 

定義から考えていく。

 

\(\nabla r=\biggl(\displaystyle\frac{\partial r}{\partial x} , \displaystyle\frac{\partial r}{\partial y} , \displaystyle\frac{\partial r}{\partial z}\biggr)\)

 

\(\displaystyle\frac{\partial r}{\partial x}\)\(=\displaystyle\frac{\partial }{\partial x}(x^2+y^2+z^2)^{\frac{1}{2}}=\displaystyle\frac{1}{2}(x^2+y^2+z^2)^{-\frac{1}{2}}\cdot 2x=\displaystyle\frac{x}{\sqrt{x^2+y^2+z^2}}=\)\(\displaystyle\frac{x}{r}\)

 

同様に、 \(\displaystyle\frac{\partial r}{\partial y}=\displaystyle\frac{y}{r}\)および\(\displaystyle\frac{\partial r}{\partial z}=\displaystyle\frac{z}{r}\) なので

 

\(\nabla r=\biggl(\displaystyle\frac{\partial r}{\partial x} , \displaystyle\frac{\partial r}{\partial y} , \displaystyle\frac{\partial r}{\partial z}\biggr)=\biggl(\displaystyle\frac{x}{r} , \displaystyle\frac{y}{r} , \displaystyle\frac{z}{r}\biggr)=\displaystyle\frac{\boldsymbol{r}}{r}\)

 

※ \(\boldsymbol{r}=(x , y , z)\) である。

 

2番

\(\nabla \displaystyle\frac{1}{r}=\biggl[\displaystyle\frac{\partial }{\partial x}\biggl(\displaystyle\frac{1}{r}\biggr) , \displaystyle\frac{\partial }{\partial y}\biggl(\displaystyle\frac{1}{r}\biggr) , \displaystyle\frac{\partial }{\partial z}\biggl(\displaystyle\frac{1}{r}\biggr)\biggr]\)

 

\(\displaystyle\frac{\partial }{\partial x}\biggl(\displaystyle\frac{1}{r}\biggr)\)\(=\displaystyle\frac{\partial }{\partial x}(x^2+y^2+z^2)^{-\frac{1}{2}}=-\displaystyle\frac{1}{2}(x^2+y^2+z^2)^{-\frac{3}{2}}\cdot 2x\)

 

\(=-\displaystyle\frac{x}{(x^2+y^2+z^2)^{\frac{3}{2}}}=\)\(-\displaystyle\frac{x}{r^3}\)

 

同様に、 \(\displaystyle\frac{\partial }{\partial y}\biggl(\displaystyle\frac{1}{r}\biggr)=-\displaystyle\frac{y}{r^3}\)および\(\displaystyle\frac{\partial }{\partial z}\biggl(\displaystyle\frac{1}{r}\biggr)=-\displaystyle\frac{z}{r^3}\) なので

 

\(\nabla \displaystyle\frac{1}{r}=\biggl[\displaystyle\frac{\partial }{\partial x}\biggl(\displaystyle\frac{1}{r}\biggr) , \displaystyle\frac{\partial }{\partial y}\biggl(\displaystyle\frac{1}{r}\biggr) , \displaystyle\frac{\partial }{\partial z}\biggl(\displaystyle\frac{1}{r}\biggr)\biggr]=\biggl(-\displaystyle\frac{x}{r^3} , -\displaystyle\frac{y}{r^3} , -\displaystyle\frac{z}{r^3}\biggr)=-\displaystyle\frac{\boldsymbol{r}}{r^3}\)

 

3番 

\(\nabla r^n=\biggl(\displaystyle\frac{\partial (r^n)}{\partial x} , \displaystyle\frac{\partial (r^n)}{\partial y} , \displaystyle\frac{\partial (r^n)}{\partial z}\biggr)\)

 

\(\displaystyle\frac{\partial (r^n)}{\partial x}\)\(=\displaystyle\frac{\partial }{\partial x}(x^2+y^2+z^2)^{\frac{n}{2}}=\displaystyle\frac{n}{2}(x^2+y^2+z^2)^{\frac{n}{2}-1}\cdot 2x=\)\(nxr^{n-2}\)

 

同様に、 \(\displaystyle\frac{\partial (r^n)}{\partial y}=ny r^{n-2}\)および\(\displaystyle\frac{\partial (r^n)}{\partial z}=nzr^{n-2}\) なので

 

\(\nabla r^n=\biggl(\displaystyle\frac{\partial r^n}{\partial x} , \displaystyle\frac{\partial r^n}{\partial y} , \displaystyle\frac{\partial r^n}{\partial z}\biggr)=(nxr^{n-2} , nyr^{n-2} ,nzr^{n-2})=nr^{n-2}\boldsymbol{r}\)

 

 

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