보존 벡터장

Let $\varphi$ be a potential for $\mathbf{F}$. Define $\phi=\varphi+ax+by+cz$. Then ${ \boldsymbol{\nabla} } \phi = { \boldsymbol{\nabla} }\varphi+(a,b,c)=\mathbf{F}+(a,b,c)$. So, $\mathbf{F}+(a,b,c)$ is also conservative.

a. If $\mathbf{F}+\mathbf{G}$ is conservative for any particular $\mathbf{F}$ and $\mathbf{G}$, then by definition, there exists a potential $\varphi$ with $\mathbf{F}+\mathbf{G} = { \boldsymbol{\nabla} } \varphi$.

Since $\mathbf{F}$ is conservative, there also exists a potential $\psi$ with $\mathbf{F} = { \boldsymbol{\nabla} } \psi$.

But now $\mathbf{G} = (\mathbf{F}+\mathbf{G})-\mathbf{F}={ \boldsymbol{\nabla} } \varphi - { \boldsymbol{\nabla} } \psi = { \boldsymbol{\nabla} }(\varphi-\psi)$. That means the function $(\varphi-\psi)$ is a potential for $G$. However, this is impossible: since $\mathbf{G}$ is non-conservative, no function with this property exists.

So it is not possible that $\mathbf{F}+\mathbf{G}$ is conservative. It must be non-conservative. b. Counterexample: if $\mathbf{F} = -\mathbf{G}$, then $\mathbf{F}+\mathbf{G} = \mathbf 0 = { \boldsymbol{\nabla} } c$ for any constant $c$. c. Since both fields are conservative, they both have potentials, say $\mathbf{F}={ \boldsymbol{\nabla} } \varphi$ and $\mathbf{G} = { \boldsymbol{\nabla} } \psi$. Then $\mathbf{F}+\mathbf{G} = { \boldsymbol{\nabla} }\varphi+{ \boldsymbol{\nabla} }\psi={ \boldsymbol{\nabla} }(\varphi+\psi)$. That is, $(\varphi+\psi)$ is a potential for $\mathbf{F}+\mathbf{G}$, so $\mathbf{F}+\mathbf{G}$ is conservative.