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anonymous
 one year ago
ques
anonymous
 one year ago
ques

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Let \[\phi(x,y)\] define a scalar field then will \[\frac{\partial^2\phi}{\partial x\partial y}\] and \[\frac{\partial^2 \phi}{\partial y \partial x}\] always be continuous? Also if \[\frac{\partial \phi}{\partial x}=0\] then \[\implies \frac{\partial^2 \phi}{\partial x \partial y}=\frac{\partial}{\partial x}(\frac{\partial \phi}{\partial y})=0?\]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0Essentially, you're asking if the mixed partials are always equal ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I know they are not always, they must be continuous, but is it true for all cases when we talk about a scalar field?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I was thinking about the different ambiguities that arise when we say the following determinant is 0 \[\left[\begin{vmatrix}i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y}& \frac{\partial}{\partial z}\\ \frac{\partial \phi}{\partial x} & \frac{\partial \phi}{\partial y} & \frac{\partial \phi}{\partial z}\end{vmatrix}\right]\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0and it IS 0 that's an identity

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0Curl of gradient is 0, yeah

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0How can it be 0 if we are ambigious about for example the expression \[\frac{\partial^2 \phi}{\partial y \partial z}\frac{\partial^2 \phi}{\partial z \partial y}\] This is not always 0 yet the curl of gradient is 0 vector

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0Clearly \(\phi \) is the potential function of vector field \(F=\nabla \phi\) Since a potential function exists, the field \(F\) is conservative by definition. Equivalently \(\nabla \times F\) is 0.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0wait a second, i think i got your question you're saying the mixed partials are "required" to be equal for the curl to be 0 ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yep that's what im saying

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0at least when we talk about a scalar field, not talking about other multivariable functions

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0The individual components need to be 0 so that curl(grad phi) is a null vector

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0Rigt, i get it... I don't seem to have a convincing explanation for myself.. let me think

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I guess I'll doze off for a few hours, really tired. I will check this out later if you have found an explanation

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0let me tag few others @eliassaab @zzr0ck3r @SithsAndGiggles @Michele_Laino @Empty @UnkleRhaukus @oldrin.bataku @Astrophysics

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0@mukushla @adxpoi @jtvatsim

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Ok, so I'm back, not much luck huh lol

imqwerty
 one year ago
Best ResponseYou've already chosen the best response.0OO i jst know that we can use curl method to check if a force is conservative or not.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0@ganeshie8 Looks like I've the found the solution to this, apparently in physics, we assume our functions to be "nice" http://mathworld.wolfram.com/PartialDerivative.html

Empty
 one year ago
Best ResponseYou've already chosen the best response.0I had some problems when I asked a similar question, you might enjoy because it wasn't given a satisfactory answer I feel. http://math.stackexchange.com/questions/956095/whendoesorderofpartialderivativesmatter
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