Ok here goes....
Find:
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this seems right to me (though i'm not sure if i'm supposed to use a chain rule), it's the next ones i'm not sure about....
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I've been at this one for hours, i think i can use the following, but i'm getting nowhere.
so....
can i divide through by and then work it all out to get
?
please help, i don't have a clue what i'm doing!!!
This is a very strange question, and I'd like to be sure that I understand the notation before I try to answer it.
I'm guessing that those little subscripts indicate a quantity that is to be held constant. If so, then , for example, makes good sense: z is a function of x and y, and when you take the partial derivative of z with respect to x you would naturally interpret this to mean that y is being held constant. But the next derivative, , is another matter altogether. The variables r and ϑ go together, and when you form a partial derivative with respect to ϑ you would normally expect that r, not x, is the variable being held constant. To keep x constant when differentiating with respect to ϑ is unusual, not to say perverse. It can be done, but it needs to be done with care. Am I right in thinking that this is what this problem is really about, or am I misinterpreting it?
yer i have copied the question exactly and correctly so it's probably just unusual but not necessarily wrong.
For the last one I get:
yer and the subscripts are the constants
hmm, your guess is better than mine, though i think it's right, the question isn't likely to be wrong at least
Okay, for , I would interpret this as meaning that r and ϑ are variables, and x is constant. (But r and ϑ are not independent variables, because they are constrained by the fact that x has to be held constant.)
Differentiate with respect to ϑ, using the chain rule and treating x as a constant: . Now we have to find .
Differentiate with respect to ϑ, using the product rule and the chain rule: .
Finally, we have to find . For this, differentiate the equation with respect to ϑ (remembering that x is constant): .
Solve that last equation for , substitute it into the equation for , and then substitute that into the equation for . When you put it all together, you should find that . Since r and ϑ are the variables, it's probably best to express this in terms of them, namely (after a bit of simplification) .
(Note: all the above partial derivatives should really have have that little subscript _x tacked on to them, just to emphasise that x is being held constant throughout.)
You could find the other derivative by exactly the same sort of method, this time keeping y fixed throughout. In practice, however, it would be much simpler to notice that , so , regardless of what else is being kept constant.