Calculus D: Partial Differentiation

D1. Surfaces

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D3. Error approximation

The acceleration of gravity can be found from the length $l$ and period $T$ of a pendulum; the formula is $g = 4\pi^2 l/T^2$. Using the linear approximation, find the relative error in $g$ (i.e. $\Delta g/g$) in the worst case if the relative error in $l$ is $5\%$ and the relative error in $T$ is $2\%$.

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D6. Exact differentials

Prove that given a general relation $f(P, V, T) = 0$ between $P, V,$ and $T$, then we have \begin{align*} \left(\frac{\partial P}{\partial V}\right)_T \left(\frac{\partial V}{\partial T}\right)_P \left(\frac{\partial T}{\partial P}\right)_V &= -1, \\ \left(\frac{\partial P}{\partial V}\right)_T = \frac{1}{\left(\frac{\partial V}{\partial P}\right)_T} \end{align*}

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D9. Stationary Points

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D10. Exact differentials

a) Which of the following are exact differentials? For those that are exact, find $f$: \begin{align*} \de{f} &= x\de{y} + y\de{x} \\ \de{f} &= x\de{y} - y\de{x} \\ \de{f} &= x\de{x} + y\de{y} + z\de{z}. \end{align*}

b) What is the value of $\oint x\de{y} + y\de{x}$ around the curve $x^4 + y^4 = 1$?

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