Topic 5 - Functions of Several Variables

Week 15 Material

Question 5.1 \(\;\) Calculate \(\partial f/\partial x\) and \(\partial f/\partial y\) when \(f(x,y)\) is given by:

\(\;(a)\;x^2+y^2\,\sin (xy),\hspace{1cm}\) \(\;(b)\;(x+y)/(x-y),\hspace{1cm}\) \(\;(c)\;\sqrt{x^2+y^2},\hspace{1cm}\) \(\;(d)\;(x^2+y^2)^{-1/2},\)

\(\;(e)\;x^y,\hspace{1cm}\) \(\;(f)\;\ln (x^2+y^2),\hspace{1cm}\) \(\;(g)\;xy+x^3\cos(xy),\hspace{1cm}\) \(\;(h)\;xy/(x+y).\)

Question 5.2 \(\;\) Calculate \(\partial f/\partial x\), \(\partial f/\partial y\) and \(\partial f/\partial z\) when \(f(x,y,z)\) is:

\(\;(a)\;xy^3-yz^2,\hspace{1cm}\) \(\;(b)\;z\left({\dfrac{x-y}{x+y}}\right),\hspace{1cm}\) \(\;(c)\;x^yz,\hspace{1cm}\) \(\;(d)\;x\cos (yz).\)

Question 5.3 \(\;\) Check that \(\partial^2 f/\partial x\partial y\) and \(\partial^2 f/\partial y\partial x\) are equal when \(f(x,y)\) is given by:

\(\;(a)\;x^2y^3+e^x+\ln y,\hspace{1cm}\) \(\;(b)^*\;x\tan\left(y^2\right),\hspace{1cm}\) \(\;(c)\;\tan^{-1}(y/x),\hspace{1cm}\) \(\;(d)\;(x^2+y^2)^{-1/2}.\)

Question 5.4 \(\;\) Find a value of the constant \(n\) such that \(v(r,t)=t^ne^{-r^2/4t}\) satisfies \[\frac{\partial v}{\partial t} = \frac{1}{r^2}\frac{\partial}{\partial r} \left(r^2\frac{\partial v}{\partial r}\right).\]

Question 5.5 \(\;\) Let \(f(x,y)= u(x,y)e^{ax+by}\) where \(u(x,y)\) satisfies \(\partial^2u/\partial x\partial y=0\). Find values of \(a\) and \(b\) such that \[\frac{\partial^2f}{\partial x\partial y}- \frac{\partial f}{\partial x}-\frac{\partial f}{\partial y}+f=0.\]

Question 5.6 \(\!{}^*\;\) Laplace’s equation \(\;\dfrac{\partial^2 f}{\partial x^2}+\dfrac{\partial^2 f}{\partial y^2}=0\;\) is of fundamental importance in physics. Find all solutions of the form \[f(x,y)=ax^3+bx^2y+cxy^2+dy^3\] where \(a\), \(b\), \(c\) and \(d\) are constants.

Question 5.7 \(\;\) The temperature at a point \((x,y,z)\) is given by \(T(x,y,z) = \lambda \sqrt{x^2+y^2+z^2}\) where \(\lambda\) is a constant. Use the chain rule to find the rate of change of temperature with respect to \(t\) along the helix path \(\left(x(t),y(t),z(t)\right)=\left(\cos t,\sin t,t\right)\). (Give your answer in terms of \(t\).)

Question 5.8 \(\;\) For the following functions, find \(df/dt\) by using the chain rule

\(\;(a)\;\;f(x,y)=x^2+y^2\) and \(x(t)=\cos t\), \(y(t)=\sin t\).

\(\;(b)\;\;f(x,y)=x^2-y^2\) and \(x(t)=\cos t + \sin t\), \(y(t)=\cos t - \sin t\).

\(\;(c)^*\;\;f(x,y,z)=x/z+y/z\) and \(x(t)=\cos^2 t\), \(y(t)=\sin^2 t\), \(z(t)=1/t\).

Check your answers by finding \(f(t)\) in terms of \(t\) explicitly and then differentiating with respect to \(t\).

Question 5.9 \(\;\) Let \(f(x,y,z)=x^2e^{2y}\cos 3z\). Find the value of \(df/dt\) at the point \((1,\ln 2,0)\) on the curve \(x=\cos t\), \(y=\ln(t+2)\), \(z=t\).

Question 5.10 \(\;\) Let \(f(x,y)=xe^{\sin y}\) where \(x=\sin\left(u^2+v^2\right)\) and \(y=u^2+v^2\). Using an appropriate form of the chain rule, find \(\partial f/\partial u\) as a function of \(u\) and \(v\). Verify your result by substituting to find \(f(u,v)\) directly in terms of \(u\) and \(v\), followed by differentiating with respect to \(u\).

Question 5.11 \(\;\) If \(f(x,y,z)\) is a function of \(x=s^3+t^2\), \(y=s^2-t^3\), \(z=s^2t^3\), express \(\partial f/\partial s\) and \(\partial f/\partial t\) in terms of \(\partial f/\partial x\), \(\partial f/\partial y\), \(\partial f/\partial z\), \(s\) and \(t\).

Question 5.12 \(\;\) If \(f(x,y)\) is a function of \(x=e^u \cos v\) and \(y=e^u \sin v\), express \(\partial f/\partial u\) and \(\partial f/\partial v\) in terms of \(\partial f/\partial x\) and \(\partial f/\partial y\). Furthermore, show that \[e^{-2u}\left(\frac{\partial^2f}{\partial u^2}+\frac{\partial^2f}{\partial v^2}\right) =\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}.\]

Question 5.13 \(\;\) Let \(f(r)\) be a function of \(r=\sqrt{x^2+y^2}\). Show that \[\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2} =\frac{d^2f}{d r^2}+\frac{1}{r}\frac{d f}{d r}.\]

Question 5.14 \(\;\) Find the gradient \(\nabla f\) for the following functions:

\(\;(a)\;\;f(x,y)=e^x\cos y,\hspace{3.2cm}\) \(\;(b)\;\;f(x,y)=x\sin{x}\cos{y},\)

\(\;(c)\;\;f(x,y,z)=\ln (x^2+2y^2 -3z^2),\hspace{2em}\) \(\;(d)\;\;f(x,y,z)=\sin{x}\cos{y}\tan{z}.\)

Question 5.15 \(\;\) Find the directional derivative of

\(\;(a)\;\;f(x,y,z)=x^2+2y^2+3z^2\) at \((1,1,0)\) in the direction \({\bf i}-{\bf j}+2{\bf k}\).

\(\;(b)^*\;\;f(x,y,z)=x\sin{y}+2z^3\) at \(\left(1,0,-1\right)\) in the direction \({\bf i}+{\bf j}\).

\(\;(c)\;\;f(x,y,z)=x+y^2+xz^2\) at \((1,2,-1)\) in the direction \({\bf i}+{\bf j}-3{\bf k}\).

\(\;(d)\;\;f(x,y,z)=xye^{x^2+z^2-5}\) at \(\left(1,3,-2\right)\) in the direction \(3{\bf i}-{\bf j}+4{\bf k}\).

Question 5.16 \(\;\) At the point \({\bf p}=(1,-1,2)\), a function \(f(x,y,z)\) has directional derivative equal to \(D_1=-\sqrt{2}\) in the direction \({\bf i}+{\bf k}\), \(D_2=1/\sqrt{2}\) in the direction \({\bf j}+{\bf k}\) and \(D_3=0\) in the direction \({\bf i}+{\bf j}+{\bf k}\). Determine \(\nabla f({\bf p})\), and calculate the directional derivative at \({\bf p}\) in the direction \({\bf i}-2{\bf j}+3{\bf k}\).

Question 5.17 \(\;\) Let \(f(x,y,z)=axy^2+byz+cz^2x^3\). Find values of the constants \(a\), \(b\) and \(c\) such that at the point \((1,2,-1)\) the directional derivative of \(f\) takes its maximum value in the direction of the positive \(z\)-axis and that value is 64.

Question 5.18 \(\;\) The temperature at each point \((x,y)\) of a metal plate is \(T(x,y)=1+x^2-y^2\). A cold ant walks in a curve on the plate, always moving in the direction where the temperature increases fastest.

\(\;(a)\;\) By considering \(\nabla T\), explain why we may choose the coordinates \(\left(x(t),y(t)\right)\) at time \(t\) to satisfy \[\dot{x}(t)=2x(t)\quad\text{and}\quad\dot{y}(t)=-2y(t).\]

\(\;(b)\;\) The ant begins at the point \((-2,1)\). Solve these two differential equations and show that the ant moves along the hyperbola \(xy=-2\).

Week 16 Material

Question 5.19 \(\;\) Find the constant \(C\) so that the level surface \(f(x,y,z)=C\) contains the point \(P\) where:

\(\;(a)\;\;f(x,y,z)=x^2+y^2+z^2\) and \(P=(1,2,3)\),

\(\;(b)\;\;f(x,y,z)=\sqrt{x^2+y^2}-\ln{z}\) and \(P=(3,4,e)\).

Question 5.20 \(\;\) Write down the equations of the tangent plane and the normal line to the surface \(x^2+2yz=2\) at the point \((a,b,c)\). Find the equations of the tangent planes to this surface which are parallel to the plane \(4x+y-7z=0\). Find also the co-ordinates of the point where the normal at \((2,1,-1)\) meets the surface again.

Question 5.21 \(\;\) Let \(S\) be the surface with equation \(xy+2yz+3zx=0\) and let \(P\) be the point \((1,-1,1)\) on the surface. Find Cartesian equations of the normal line and the tangent plane to \(S\) at \(P\). Find the point where the normal at \(P\) meets \(S\) again.

Question 5.22 \(\;\) Find a unit normal vector to the surface \(2x^3z+x^2y^2+xyz-4=0\) at the point \((2,1,0)\).

Question 5.23 \(\;\) Find a vector \({\bf v}\) in terms of \(x\), \(y\) and \(z\) which is normal to the surface \[z=\sqrt{x^2+y^2}+(x^2+y^2)^{3/2}\] at a general point \((x,y,z) \not= (0,0,0)\) of the surface. Find the cosine of the angle \(\theta\) between \({\bf v}\) and the \(z\)-axis and determine the limit of \(\cos \theta\) as \((x,y,z)\to (0,0,0)\).

Question 5.24 \(\;\) Evaluate the divergence \(\nabla\cdot{\bf A}\) and curl \(\nabla\times{\bf A}\) for the following vector fields at the given points

\(\;(a)\;\;{\bf A}({\bf x})=(x^2+y^2+z^2,yz+zx+xy,x^2y^2z^2)\) at \((1,2,-1)\).

\(\;(b)\;\;{\bf A}({\bf x})=\left(xy^2-z\cos^2 x,x^2\sin y,xyz\right)\) at \(\left(\pi/4,\pi/4,-1\right)\).

\(\;(c)^*\;\;{\bf A}({\bf x})=(\cos x,\cos y,\cos z)\) at \(\left(\pi/4,\pi/3,\pi/2\right)\).

\(\;(d)\;\;{\bf A}({\bf x})=\left(\cos{y},\cos{z},\cos{x}\right)\) at \(\left(\pi/4,\pi/3,\pi/2\right)\).

Question 5.25 \(\;\) For each of the following functions, find all the critical points and classify each as a local maximum, local minimum or saddle point.

\(\;(a)\;\;x^2 +y^4-2x-4y^2+5,\hspace{2cm}\) \(\;(b)\;\;2x^3-9x^2y+12xy^2-60y,\)

\(\;(c)^*\;\;(x^2+y^2)^2-8(x^2-y^2),\hspace{2.cm}\) \(\;(d)\;\;x^2y^2-x^2-y^2,\)

\(\;(e)\;\;y^2+xy+x^2+4y-4x+5,\hspace{1cm}\) \(\;(f)\;\;y^2+\sin x,\)

\(\;(g)^*\;\;x^2-\sin y,\hspace{2.9cm}\) \(\;(h)\;\;\cos x\sin y,\hspace{3.35cm}\) \(\;(i)\;\;\sin x\sin y.\)

Question 5.26 \(\;\) On a map of the Lake District, the height of a point with coordinates \((x,y)\) is \(h(x,y)=125y^2-100x^2+50xy+400\) metres above sea level. Find the direction and rate of fastest ascent when at the point with coordinates \((2,1)\) and walking at one km per hour. Also find the directions in which one can traverse the slope, i.e. walk horizontally.