398
Evaluate the integral of the function \(f(x,y,z) = x+y\) over the surface \(S\) given by: \[ \Phi(u,v) = ( 2u \cos v, 2u \sin v, u ), \quad u \in [0,4], v \in [0, \pi] \]
Evaluate the integral of the function \(f(x,y,z) = z+6\) over the surface \(S\) given by: \[ \Phi(u,v) = ( u, \frac{v}{3}, v ), \quad u \in [0,2], v \in [0, 3]. \]
Evaluate the integral \[ {\int\!\!\!\int}_{S} (3x - 2y + z) \,dS, \] where \(S\) is the portion of the plane \(2x + 3y + z = 6\) that lies in the first octant.
Evaluate the integral \[ {\int\!\!\!\int}_{S} (x+z) \,dS, \] where \(S\) is the part of the cylinder \(y^2 + z^2 =4\) with \(x \in [0,5]\).
Let \(S\) be the surface defined by \[ \Phi(u,v) = ( u+v, u-v, uv ). \]
Evaluate the integral \[ {\int\!\!\!\int}_{S} (x^{2}z + y^{2}z) \,dS, \] where \(S\) is the part of the plane \(z=4+x+y\) that lies inside the cylinder \(x^{2}+y^{2}=4\).
Compute \({\int\!\!\!\int}_S {\it xy} \,dS\), where \(S\) is the surface of the tetrahedron with sides \(z=0, y=0\), \(x + z = 1\), and \(x = y\).
Evaluate \({\int\!\!\!\int}_S {\it xyz} \,dS\), where \(S\) is the triangle with vertices \((1,0, 0), (0, 2, 0)\), and \((0, 1, 1)\).
Evaluate \({\int\!\!\!\int}_S z \,dS\), where \(S\) is the upper hemisphere of radius \(a\), that is, the set of \((x, y, z)\) with \(z = \sqrt{a^2 - x^2 - y^2}\).
Evaluate \({\int\!\!\!\int}_S (x+y+z)\,dS\), where \(S\) is the boundary of the unit ball \(B \); that is, \(S\) is the set of \((x, y, z)\) with \(x^2 + y^2 + z^2 = 1\). (HINT: Use the symmetry of the problem.)
Verify that in spherical coordinates, on a sphere of radius R, \[ \|{\bf T}_\phi\times {\bf T}_\theta \|\, d\phi\, d\theta=R^2\sin\phi\ d\phi\ d\theta. \]
Evaluate \({\int\!\!\!\int}_S z \,dS\), where \(S\) is the surface \(z = x^2 + y^2, x^2 + y^2 \le 1\).
Evaluate the surface integral \({\int\!\!\!\int}_S z^2 \,dS\), where \(S\) is the boundary of the cube \(C=[-1,1]\times [-1,1] \times [-1,1]\). (HINT: Do each face separately and add the results.)
Find the mass of a spherical surface \(S\) of radius \(R\) such that at each point \((x, y, z) \in S\) the mass density is equal to the distance of \((x, y, z)\) to some fixed point \((x_0, y_0, z_0)\in S\).
A metallic surface \(S\) is in the shape of a hemisphere \(z \,{=}\, \sqrt{R^2 \,{-}\,x^2 \,{-}\, y^2}\), where \((x, y)\) satisfies \(0 \le x^2 \,{+}\,y^2 \le R^2\). The mass density at \((x, y, z)\in S\) is given by \(m(x,y,z) = x^2 + y^2\). Find the total mass of \(S\).
Let \(S\) be the sphere of radius \(R\).
399
Find the average value of \(f(x,y,z) = x + z^2\) on the truncated cone \(z^{2} = x^{2}+y^{2}\), with \(1 \leq z \leq 4\).
Evaluate the integral \[ {\int\!\!\!\int}_{S} (1-z) \,dS, \] where \(S\) is the graph of \(z=1 - x^{2} - y^{2}\), with \(x^{2}+y^{2} \leq 1\).
Find the \(x, y\), and \(z\) coordinates of the center of gravity of the octant of the solid sphere of radius \(R\) and centered at the origin determined by \(x \ge 0 , y \ge 0, z \ge 0\). (HINT: Write this octant as a parametrized surface—see Example 3 of this section and Exercise 18.)
Find the \(z\) coordinate of the center of gravity (the average \(z\) coordinate) of the surface of a hemisphere \((z \le 0)\) with radius \(r\) (see Exercise 18). Argue by symmetry that the average \(x\) and \(y\) coordinates are both zero.
Let \({\Phi}{:}\, D \subset {\mathbb R}^{2} \to {\mathbb R}^{3}\) be a parametrization of a surface \(S\) defined by \[ x=x(u, v),\qquad y=y(u, v),\qquad z=z(u, v). \]
Dirichlet’s functional for a parametrized surface \({\Phi}\colon\, D\to {\mathbb R}^3\) is defined byfootnote # \[ J({\Phi}) = \frac{1}{2} \int\!\!\!\int\nolimits_{D} \Big(\Big\|\frac{\partial{\Phi}}{\partial u}\Big\|^2+ \Big\|\frac{\partial{\Phi}}{\partial v}\Big\|^2\Big)\, du \,dv. \]
Use Exercise 23 to argue that the area \(A({\Phi}) \le J({\Phi})\) and equality holds if \[ \hbox{(a)} \Big\|\frac{\partial {\Phi}}{\partial u}\Big\|^2 = \Big\|\frac{\partial{\Phi}}{\partial v}\Big\|^2\qquad \hbox{and }\qquad \hbox{(b)} \frac{\partial{\Phi}}{\partial u}\, {\cdot}\, \frac{\partial{\Phi}}{\partial v} = 0. \]
Compare these equations with Exercise 23 and the remarks at the end of Section 7.5. A parametrization \({\Phi}\) that satisfies conditions (a) and (b) is said to be conformal.
Let \(D\subset{\mathbb R}^2\) and \({\Phi}\colon\, D \to {\mathbb R}^2\) be a smooth function \({\Phi}(u,v) = (x(u,v),y(u,v))\) satisfying conditions (a) and (b) of Exercise 16 and assume that \begin{eqnarray*} \hbox{ det } \left[\begin{array}{l@{\qquad}l} \\[-8pt] \displaystyle\frac{\partial x}{\partial u} & \displaystyle\frac{\partial x}{\partial v}\\[11pt] \displaystyle\frac{\partial y}{\partial u} &\displaystyle\frac{\partial y}{\partial v}\\[7pt] \end{array}\right] >0.\\[-4pt] \end{eqnarray*}
Show that \(x\) and \(y\) satisfy the Cauchy–Riemann equations \(\partial x/\partial u = \partial y/\partial v, \partial x/\partial v = - \partial y/\partial u\).
Conclude that \(\nabla^2{\Phi} = 0\) (i.e., each component of \({\Phi}\) is harmonic).
Let \(S\) be a sphere of radius \(r\) and \({\bf p}\) be a point inside or outside the sphere (but not on it). Show that \[ \int\!\!\!\int\nolimits_{S}\frac{1}{\|{\bf x}-{\bf p}\|} \,dS = \left\{ \begin{array}{l@{\qquad}l} 4\pi r &\hbox{if}\quad{\bf p}\hbox{ is inside } S\\ 4\pi r^2/d &\hbox{if}\quad{\bf p}\hbox{ is outside } S, \end{array}\right. \] where \(d\) is the distance from \({\bf p}\) to the center of the sphere and the integration is over the sphere. [HINT: Assume \({\bf p}\) is on the \(z\)-axis. Then change variables and evaluate. Why is this assumption on \({\bf p}\) justified?]
400
Find the surface area of that part of the cylinder \(x^2 + z^2 = a^2\) that is inside the cylinder \(x^2 + y^2 = 2ay\) and also in the positive octant \((x \ge 0, y \ge 0, z \ge 0)\). Assume \(a > 0\).
Let a surface \(S\) be defined implicitly by \(F(x, y, z) = 0\) for \((x, y)\) in a domain \(D\) of \({\mathbb R}^2\). Show that \begin{eqnarray*} &&\int\!\!\!\int\nolimits_{S} \Big|\frac{\partial F}{\partial z}\Big|\,dS \\[4pt] &&= \int\!\!\!\int\nolimits_{D} \sqrt{\Big(\frac{\partial F}{\partial x}\Big)^2 + \Big(\frac{\partial F}{\partial y}\Big)^2 + \Big(\frac{\partial F}{\partial z}\Big)^2} \,{\it dx} \,{\it dy}. \end{eqnarray*}
Compare with Exercise 22 of Section 7.5.