Both conservative and solenoidal vector field

aruna1
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Homework Statement



if a vecor A is both solenoidal and conservative; is it correct that

A=-▼Φ

that is

A=- gradΦ

Φ is a scalar function

thanks
 
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If A is just a conservative vector field, then A= -\nabla \phi for some scalar function \phi. I'm not sure what requiring that it also be solenoidal adds.
 
HallsofIvy said:
If A is just a conservative vector field, then A= -\nabla \phi for some scalar function \phi. I'm not sure what requiring that it also be solenoidal adds.

well i thouhgt if A is only conservative
then

A= \nabla \phi (according to my textbook)

not

A= -\nabla \phi

so what is right?:confused:
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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