Multivariable Calculus: Problem Set 6

1. Equality of the mixed derivatives

Let \[ f(x, y) = \frac{xy(x^2-y^2)}{x^2 + y^2}, \]

away from $(0,0)$ and zero otherwise. Show that \[ \pd{^2 f}{y\partial x} \neq \pd{^2 f}{x\partial y} \]

at the origin.

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2. Vector identities

Part a (sum proof)

For vector fields $\mathbf{F}$, $\mathbf{G}$, show that \[ \nabla \cdot (\mathbf{F} \times \mathbf{G}) = \mathbf{G} \cdot (\nabla \times \mathbf{F}) - \mathbf{F}\cdot (\nabla \times \mathbf{G}). \]

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Part a (index notation)

Prove the same identity using index notation.

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Part b

Show also that \[ \nabla \times (\mathbf{F} \times \mathbf{G}) = \mathbf{F}(\nabla \cdot \mathbf{G}) - \mathbf{G}(\nabla \cdot \mathbf{F}) + (\mathbf{G} \cdot \nabla)\mathbf{F} - (\mathbf{F} \cdot \nabla)\mathbf{G}. \]

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Part b (index notation)

Prove the same identity using index notation.

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3. Change of coordinates

Let $\phi(x,y,z) = y^2 - xz$ and $\mathbf{f}(x,y,z) = (z^2, x^2, y^2)$.

a) Find $\nabla \phi$, $\nabla \times \mathbf{f}$, and $\nabla \cdot \mathbf{f}$.

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b) For the orthogonal basis $\mathbf{e}_1 = [0, -1, 0]$, $\mathbf{e}_2 = [1, 0, -1]/\sqrt{2}$, $\mb{e}_3 = [1,0,1]/\sqrt{2}$, create new coordinates $X$, $Y$, $Z$, such that \[ X \mb{e}_1 + Y\mb{e}_2 + Z\mb{e}_3 = x\mb{i} + y\mb{j} + z\mb{k}. \]

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c) Determine $x$, $y$, $z$ in terms of $X$, $Y$, $Z$. Find also $\Phi$, $F_1$, $F_2$, and $F_3$ such that $\Phi(X,Y,Z) = \phi(x,y,z)$ and $F_1 \mb{e}_1 + F_2 \mb{e}_2 + F_3 \mb{e}_3 = f_1\mb{i} + f_2\mb{j} + f_3\mb{k}$.

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d) Verify by direct calculation that the formulae for grad, div, and curl, are irrespective of what right handed orthonormal co-ordinate system is used.

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4. Divergence for polar coordinates

Let $r$ and $\theta$ denote plane polar coordinates and set $\mb{e}_r = [\cos\theta, \sin\theta, 0]$, and $\mb{e}_\theta = [-\sin\theta, \cos\theta, 0]$. Let $\mb{F}(r,\theta) = F_r \mb{e}_r + F_\theta \mb{e}_\theta$ be a vector field. Prove that \[ \nabla \cdot \mathbf{F} = \frac{1}{r}\pd{}{r}(rF_r) + \frac{1}{r} \pd{F_\theta}{\theta}. \]

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5. Verifying Divergence theorem

Show that \[ \iiint_R \nabla \cdot \mathbf{F} \, \de{V} = \iint_{\partial R} \mathbf{F} \cdot \de\mathbf{S}, \]

where $\mathbf{F}(x,y,z) = (F_1, F_2, F_3) \equiv (y, xy, -z)$ and $R$ is the region volume enclosed by the cylinder $x^2 + y^2 = 4$, the plane $z = 0$, and the paraboloid $z = x^2 + y^2$, and $\partial R$ is the boundary of $R$.

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