91Ó°ÊÓ

Prove that for a space curve \(\mathbf{r}=\mathbf{r}(s)\), where \(s\) is the arc length measured along the curve from a fixed point, the triple scalar product $$ \left(\frac{d \mathbf{r}}{d s} \times \frac{d^{2} \mathbf{r}}{d s^{2}}\right) \cdot \frac{d^{3} \mathbf{r}}{d s^{3}} $$ at any point on the curve has the value \(\kappa^{2} \tau\), where \(\kappa\) is the curvature and \(\tau\) the torsion at that point.

(a) Using the parameterization \(x=u \cos \phi, y=u \sin \phi, z=u \cot \Omega\), find the sloping surface area of a right circular cone of semi-angle \(\Omega\) whose base has radius \(a\). Verify that it is equal to \(\frac{1}{2} \times\) perimeter of the base \(\times\) slope height. (b) Using the same parameterization as in (a) for \(x\) and \(y\), and an appropriate choice for \(z\), find the surface area between the planes \(z=0\) and \(z=Z\) of the paraboloid of revolution \(z=\alpha\left(x^{2}+y^{2}\right)\)

Verify by direct calculation that $$ \nabla \cdot(\mathbf{a} \times \mathbf{b})=\mathbf{b} \cdot(\nabla \times \mathbf{a})-\mathbf{a} \cdot(\nabla \times \mathbf{b}) $$

Evaluate the Laplacian of the function $$ \psi(x, y, z)=\frac{z x^{2}}{x^{2}+y^{2}+z^{2}} $$ (a) directly in Cartesian coordinates, and (b) after changing to a spherical polar coordinate system. Verify that, as they must, the two methods give the same result.

(a) For cylindrical polar coordinates \(\rho, \phi, z\) evaluate the derivatives of the three unit vectors with respect to each of the coordinates, showing that only \(\partial \hat{\mathbf{e}}_{\rho} / \partial \phi\) and \(\partial \hat{\mathbf{e}}_{\phi} / \partial \phi\) are non-zero. (i) Hence evaluate \(\nabla^{2} \mathbf{a}\) when \(\mathbf{a}\) is the vector \(\hat{\mathbf{e}}_{\rho}\), i.e. a vector of unit magnitude everywhere directed radially outwards from the \(z\)-axis. (ii) Note that it is trivially obvious that \(\nabla \times \mathbf{a}=\mathbf{0}\) and hence that equation \((10.41)\) requires that \(\dot{\nabla}(\nabla \cdot \mathbf{a})=\nabla^{2} \mathbf{a}\). (iii) Evaluate \(\nabla(\nabla \cdot \mathbf{a})\) and show that the latter equation holds, but that $$ [\nabla(\nabla \cdot \mathbf{a})]_{\rho} \neq \nabla^{2} a_{\rho} $$ (b) Rework the same problem in Cartesian coordinates (where, as it happens, the algebra is more complicated).

For a description in spherical polar coordinates with axial symmetry of the flow of a very viscous fluid, the components of the velocity field \(\mathbf{u}\) are given in terms of the stream function \(\psi\) by $$ u_{r}=\frac{1}{r^{2} \sin \theta} \frac{\partial \psi}{\partial \theta}, \quad u_{\theta}=\frac{-1}{r \sin \theta} \frac{\partial \psi}{\partial r} $$ Find an explicit expression for the differential operator \(E\) defined by $$ E \psi=-(r \sin \theta)(\nabla \times \mathbf{u})_{\phi} $$ The stream function satisfies the equation of motion \(E^{2} \psi=0\) and, for the flow of a fluid past a sphere, takes the form \(\psi(r, \theta)=f(r) \sin ^{2} \theta\). Show that \(f(r)\) satisfies the (ordinary) differential equation $$ r^{4} f^{(4)}-4 r^{2} f^{\prime \prime}+8 r f^{\prime}-8 f=0 $$