Problem 1. (10 marks) Calculate the area of the region of the plane described in polar coordinates by \(0 \leq \theta \leq 1\) and \(0 \leq r \leq 1+e^{\theta}\).

Problem 2. (10 marks) Calculate \(\frac{d}{d t} f\left(g\left(t^{2}\right), g\left(t^{4}\right)\right)\) where \(f\) is a differentiable function of two variables and \(g\) is a differentiable function of one variable. Your answer should be expressed in terms of $f, g$ and their derivatives and/or partial derivatives.

Problem 3. (10 marks) Find the absolute maximum and minimum values of \(f(x, y)=x y-x-y\) on the region in the plane bounded by \(x=0, y=0\) and \(x+y=3\)

Problem 4. (10 marks) Use cylindrical coordinates to evaluate \(\iiint_{E} x^{3}+x y^{2} d V\) where \(E\) is the solid in the first octant that lies beneath the paraboloid \(z=\) \(1-x^{2}-y^{2}\). [Note: You must use cylindrical coordinates to receive any credit.]

Problem 5. ( 10 marks) Use the change of variables \(u=y-x\) and \(v=y+x\) to compute the \(\iint_{R} \cos \left(\frac{y-x}{y+x}\right) d A\) where \(R\) is the trapezoidal region with vertices \((1,0),(2,0),(0,2)\), and \((0,1)\)

Problem 6. ( 10 marks) Let \(\vec{F}\) be the vector field

\(\vec{F}(x, y, z)=\left(\frac{2 x}{x^{2}+y^{2}}, \frac{2 y}{x^{2}+y^{2}}, 1\right) .\)

Let \(S\) be the surface oriented downward and parametrised by

\(\vec{r}(u, v)=\left(u \cos (v), u \sin (v), u^{2}\right), 1 \leq u \leq 2,0 \leq v \leq 2 \pi .\) Find the surface integral \(\iint_{S} \vec{F} \cdot d \vec{S}\) by

1. using the above parametrisation of the surface \(S\) and solving the integral,
2. using the Divergence Theorem

Problem 7. (10 marks) Let \(\vec{F}\) be the vector field given by

\(\vec{F}(x, y, z)=\left(\frac{\ln \left(x^{6} y^{4} z^{2}+1\right)}{z^{4}+y^{4}+5}-y x^{2} e^{z}, \cos (\pi z) x y^{2}, e^{x^{2} y^{2}}+\sin \left(z^{3}\right)\right)\)

Let \(S\) be the upper hemisphere \((z \geq 0)\) of the spherical surface of radius 3 centred at the origin, oriented with unit outward normal. Calculate

\(\iint_{S}(\nabla \times \vec{F}) \cdot \hat{n} d S\)

Problem 8. (10 marks) Let \(\vec{F}=\left(F_{1}, F_{2}\right)\) be a continuously differentiable vector field on the region \(D\) given by the plane \(\mathbb{R}^{2}\) with the origin \((0,0)\) taken away. Assume that \(\vec{F}\) satisfies

\(\frac{\partial F_{2}}{\partial x}=\frac{\partial F_{1}}{\partial y} .\)

Let \(C_{a}\) be the circle of radius \(a\) centred at the origin, oriented counterclockwise. Show that \(\int_{C_{a}} \vec{F} \cdot d \vec{r}=K\) for some constant \(K\) independent of \(a\).

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