Exact Differentials

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by stephaniejoy19
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Exact Differentials

Exact Differentials

Exact Differentials, when talked about in the context of vectors, are sometimes called conservative fields. And since the parts that make up the exact differential are the partial derivatives of some function, they can be thought of as the components of the gradient of that function.

Hmm...looks eerily similar to the Fundamental Theorem of Calculus!

Basically, an exact differential is an expression that is made of all the partial derivatives of the SAME function. So it would look like this:Mdx + Ndy + Pdz = fxdx + fydy + fzdzThey are quite easy to integrate – that is, if you are able to tell that it is an exact differential…

So how do you tell if the expression within an integral is an exact differential?It’s actually pretty easy – recall (in two variables) that if the partial derivatives are continuous, then fxy = fyx. So, given the expression Mdx + Ndy, we know it is an exact differential IF My = Nx.In three variables: Given Mdx + Ndy + Pdz, we know it is an exact differential IF My = Nx, Mz = Px, AND Nz = Py.

Watch this example... ...And read this one!

It can be tricky to find the function, f, whose partial derivatives make up the exact differential. This function is called the potential function and the video to the left shows a method that can be used to find it. For another example of this method, watch the video here.


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