(2) (Multivariable chain rule and complex line integration) Prove the following statements. (a) Let z = u(x,y), x = x(t) and y = y(t). Then z = u(x(t), y(t)) becomes a function in single variable t. The derivative dz/dt, when it exists, can be computed by the two-variable chain rule in MATH222. Please state the chain rule, and verify the formula for the case of z(t) = (sin t)^{(cos t)} by computing z'(t) using (1) direct method, and (2) the chain rule. (Hint: a^b = e^{b ln a}.) (b) Show that if F ? H(?) and ? : [a,b] ? ? is a smooth curve, then d[(F(?(t))]/dt = F'(?(t)) ?'(t). Suggestion: Review definition! Note that ? and F ? ? are functions [a,b] ? C, while F is that of C ? C. Their derivatives are defined in different ways. (c) Let F ? H(?), w1, w2 ? ?, and f = F'. Show that the integration of f is path-independent, that is, for any piecewise smooth curve ? : [a,b] ? ? with ?(a) = w1 and ?(b) = w2, we have ?_? f(z) dz = F(w2) ? F(w1). (d) Determine, with proof, if there exists F ? H(D(0,1)) such that F'(z) = 1/z.
Added by Joaquin G.
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We have $x(t) = \sin(t)$ and $y(t) = \cos(t)$. (1) Direct method: $$z'(t) = (\sin(t) \cos(t))' = (\sin(t))' \cos(t) + \sin(t) (\cos(t))' = \cos^2(t) - \sin^2(t)$$ (2) Chain rule: $$\frac{\partial z}{\partial x} = \cos(t), \quad \frac{\partial z}{\partial y} Show more…
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