MATH115 GEOMETRY AND CALCULUS

Quiz 3

• Surface tension Consider the surface $S$ in ${ℝ}^3$ defined by the equation $f(x,y,z)=1$, where $f(x,y,z)=x^2-y+z^3$. Which of the following is not true?

  A) $(2,-1,3)$ is a normal vector to $S$ at $(1,1,1)$,$$

  B) The tangent plane at $(0,-1,0)$ is given by the equation $y=-1$,$$

  C) If $(x,y,z)$ is in $S$ then so is $(-x,y,z)$,$$

  D) The curve $\gamma :ℝ\to {ℝ}^3$, $\gamma (t)=({\sin }^{2}t\cos t,{\sin }^{4}t-1,{\sin }^{2}t)$ is contained in $S$,

  E) There exists a point $(x,y,z)$ on $S$ such that $\nabla f=(0,0,0)$.

• Staying impartial Which of the following statements about a function $f(x,y,z)$ of three variables is true?

  A) If $f_x=0$ then $f$ is constant,$$ B) If $f_{xy}=0$ then $f$ depends only on $x$ and $y$,

  C) If $f_{xy}=f_{xz}=0$ then $f_x$ is constant,$$

  D) If $f_{xyz}=0$ then $f=g(x,y)+h(x,z)+k(y,z)$ for some functions $g,h,k$.

  E) If $f_{xxx}=0$ then $f=g(y,z)+a{x}^2+bx+c$ for some function $g$ and $a,b,c\in ℝ$.

• Gradient test Which of the following vector-valued functions $𝐟=(f_1,f_2,f_3)$ can not be expressed as $\nabla \varphi$ for some $\varphi$?

  A) $𝐟=(3x^{2}y-3yz-x,x^3-3xz+y^2{z}^2,x-3xy+y^{3}z)$,$$ B) $𝐟=(\frac{1}{x+y+z},\frac{1}{x+y+z},\frac{1}{x+y+z})$,$$

  C) $𝐟=((y+x{y}^{3}z)e^{x{y}^{2}z},(x+2x^2{y}^{2}z)e^{x{y}^{2}z},x^2{y}^3{e}^{x{y}^{2}z})$,$$ D) $𝐟=(\sqrt{\frac{yz}{x}},\sqrt{\frac{xz}{y}},\sqrt{\frac{xy}{z}})$,

  E) $𝐟=(\cos x\cos y\tan z,-\sin x\sin y\tan z,\sin x\cos y{\sec }^{2}z)$.

• From one chain rule… Let $\gamma :ℝ\to {ℝ}^3$ be a parametrized curve, let $f(x,y,z)$ be a function and let $F(t)=f(\gamma (t))$. Which of the following statements is not true?

  A) If $\nabla f(\gamma (t_0))=0$, then $F^{\prime }(t_0)=0$,$$ B) If $F^{\prime }(t_0)=0$ then $\nabla f(\gamma (t_0))=0$,

  C) For any point $(x,y,z)$ the direction of the rate of greatest increase of $f$ is opposite to the direction of the rate of greatest decrease,

  D) If $F(t)$ is constant then the image of $\gamma$ lies in a surface of the form $f(x,y,z)=c$,

  E) The tangent line to $\gamma$ at $\gamma (t_0)$ is parallel to ${\gamma }^{\prime }(t_0)$.

• … to another Let $x=r\cos \theta$, $y=r\sin \theta$. Which of the following statements is not true?

  A) $\frac{\partial f}{\partial \theta }=-y\frac{\partial f}{\partial x}+x\frac{\partial f}{\partial y}$,$$ B) $\frac{\partial f}{\partial x}=\frac{\partial f}{\partial r}\cos \theta -\frac{1}{r}\frac{\partial f}{\partial \theta }\sin \theta$,$$ C) $\frac{\partial f}{\partial r}=-\frac{\partial f}{\partial x}\sin \theta +\frac{\partial f}{\partial y}\cos \theta$,

  D) If $(f_x{f}_y)=(\mathrm{0\hspace{0.33em}\hspace{0.17em}0})$ then $(f_r{f}_{\theta })=(\mathrm{0\hspace{0.33em}\hspace{0.17em}0})$,$$ E) $\frac{\partial f}{\partial y}=\frac{y}{\sqrt{x^2+y^2}}\frac{\partial f}{\partial r}+\frac{x}{x^2+y^2}\frac{\partial f}{\partial \theta }$.

MATH113 CALCULUS AND GEOMETRY

Assessed Exercises 3

• (3 marks) Find the equations of the normal line $l$ and the tangent plane to the surface $xyz=6$ at $(3,2,1)$.

  
  

• (2 marks) Show that if $f=f(x,y)$ and $f_{yy}=0$, then  $f(x,y)=g(x)y+h(x)$  for some functions $g$, $h$. (Start by stating the form taken by $f_y$.)

• (5 marks) Show that $𝒈(x,y,z)=(y{z}^2+3,x{z}^2+2z+1,\mathrm{\hspace{0.33em}2}xyz+2y)$ can be expressed as $\nabla \varphi$ for a function $\varphi$, and find $\varphi$.

• Bonus question Let and be Jacobian matrices (respectively for $x$ and $y$ in terms of $u,v$; and for $u$ and $v$ in terms of $w$, $z$).

  Show that $J_1{J}_2$ is the Jacobian matrix for $x$ and $y$ in terms of $w,z$.