1. Find the gradient $∇f$ for each of the following functions:
   (a) $f(x, y)=\mathrm e^{x^{2}-y^{2}} \sin 2 x y$
   (b) $f(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{-1 / 2}$
   Solution.
   (a) $∇f=(2\mathrm{e}^{x^2 -y^2 } {\left(y\cos \left(2xy\right)+x\sin \left(2xy\right)\right)},2\mathrm{e}^{x^2 -y^2 } {\left(x\cos \left(2xy\right)-y\sin \left(2xy\right)\right)})$
   (b) $∇f=(-x\left(x^{2}+y^{2}+z^{2}\right)^{-3/ 2},-y\left(x^{2}+y^{2}+z^{2}\right)^{-3/ 2},-z\left(x^{2}+y^{2}+z^{2}\right)^{-3/ 2})$
   
2. For each of the following functions sketch some contours $f(x, y) =$ constant (include enough contours to explore all four quadrants of $(x, y)$) and indicate $∇f$ by arrows at some points on these curves. Use one set of axes for each case.
   (a) $f(x, y)=x y$;
   (b) $f(x, y)=\frac{1}{4} x^{2}+y^{2}$;
   (c) $f(x, y)=\frac{x}{y}$.
   Solution.
   (a) $∇f=(y,x)$;
   (b) $∇f=\left(\frac{1}{2}x,2y\right)$;
   (c) $∇f=\left(\frac1y,-\frac x{y^2}\right)$.
   
3. Show that there is no function $f(x, y, z)$ such that $∇f(x, y, z) = (y, z, x)$.
   Proof.
   $\frac{\partial f}{\partial x}=y$, so $f=xy+g(y,z)$. But $x=\frac{\partial f}{\partial z}=\frac{\partial g(y,z)}{\partial z}$, no suitable $g$ exists.
   
4. Let $f(x, y) = x^2y^3$ and determine $∇f$. Given the vector $\mathbf a = (1, 1)$ and the unit vector $\mathbf v = (a, b)$ with $a^2 + b^2 = 1$, evaluate $\lim _{t \rightarrow 0} \frac{f(\mathbf{a}+t \mathbf{v})-f(\mathbf{a})}{t}$.
   What is the maximum value of this limit that can be obtained by varying $a$ in the vector $\mathbf v$?
   Solution.
   $∇f=(2xy^3,3x^2y^2,0)$.
   $\lim _{t \rightarrow 0} \frac{f(\mathbf{a}+t \mathbf{v})-f(\mathbf{a})}{t}=\lim _{t \rightarrow 0} \frac{(1+ta)^2(1+tb)^3-1}{t}=2a+3b\le\sqrt{(2^2+3^2)(a^2+b^2)}=\sqrt{13}$.
5. A bug is walking on the $xy$-plane which is has a toxic environment. The level of toxicity is given by the function $f(x, y) = 2x^2y − 3x^3$. The bug starts at the point (1,2). In what direction away from (1,2) should it initially move in order to lower its exposure to the toxin as rapidly as possible?
   Solution.
   $-∇f|_{x=1,y=2}=-(4xy-9x^2,2x^2)|_{x=1,y=2}=(1,-2)$. It should move towards the direction of 5 o'clock.
   
6. Find a unit vector perpendicular to the surface $x^2y+y^2z+z^2x + 1 = 0$ at the point (1,2,−1). Hence, or otherwise, write the equations of the tangent plane and the normal line at this point.
   Solution.
   $\nabla(x^2y+y^2z+z^2x)|_{(1, 2, −1)}=(2xy+z^2,2yz+x^2,2zx+y^2)|_{(1, 2, −1)}=(5,-3,2)$. So the equation of the normal line is $\mathbf r=(1,2,-1)+t(5,-3,2)$. The equation of the tangent plane is $5x-3y+2z=-3$.
7. Let $f$ and $g$ be sufficiently differentiable functions of $x, y, z$.
   (a) Prove the following:
   i. $∇(fg) = f∇g + g∇f$;
   ii. $∇(f^n) = nf^{n−1}∇f$;
   iii. $\nabla\left(\frac{f}{g}\right)=\frac{g \nabla f-f \nabla g}{g^{2}}$
   (b) In Cartesian coordinates $(x, y, z)$ the Laplacian operator, $∇^2$, is defined by $\nabla^{2} f=\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}+\frac{\partial^{2} f}{\partial z^{2}}$. Show that $\nabla^{2}(f g)=f \nabla^{2} g+2(\nabla f) \cdot(\nabla g)+g \nabla^{2} f$.
   Proof.
   (a)(i)$(\nabla f)_i = \dfrac{\partial f}{\partial x_i}⇒(\nabla (fg))_i = \dfrac{\partial (fg)}{\partial x_i} = \dfrac{\partial f}{\partial x_i}g + f\dfrac{\partial g}{\partial x_i} = g(\nabla f)_i + f(\nabla g)_i$ holds for each coordinate $x_i$. Hence $\nabla (fg) = g\nabla f + f \nabla g$.
   (ii)We induct on $n$. When $n=0$, we have 0=0. Assume the property is true for $n-1$, so $∇f^{n-1}=(n-1)f^{n-2}∇f$, using part (i), $∇(f^n)=∇(f^{n-1}f)= f^{n−1}∇f+f∇f^{n-1}=f^{n−1}∇f+f\cdot(n-1)f^{n-2}∇f=nf^{n−1}∇f$. By induction, it is true for all $n\in\mathbb N$.
   (iii)By (i),$\nabla f=\nabla\left(g\cdot\frac{f}{g}\right)=g\nabla\left(\frac{f}{g}\right)+\frac{f}{g}\nabla g$⇒$\nabla\left(\frac{f}{g}\right)=\frac{g \nabla f-f \nabla g}{g^{2}}$.
   (b)$\nabla^{2}(f g)=\nabla(f\nabla g+g\nabla f)=f\nabla^2g+(\nabla f)\cdot(\nabla g)+(\nabla g)\cdot(\nabla f)+g\nabla f^2=f \nabla^{2} g+2(\nabla f) \cdot(\nabla g)+g \nabla^{2} f$.
8. Use the Taylor series for a function of two variables to expand the following functions about the point $x = 1, y = 1$, to the order shown.
   (a) $f(x, y) = x^3 + y^2 + 3x^2y$, to all orders;
   (b) $f(x, y) = x^2 + y +\cos(2πxy)$, to second order.
   Solution.
   (a) $f(x, y) = x^3 + y^2 + 3x^2y$ is a polynomial. So the Taylor series is itself.
   (b) $f(1, 1) = 3,f_x=2x-2\pi y\sin\left(2\pi xy\right)⇒f_x(1,1)=2,$$f_y=1-2\pi x\sin\left(2\pi xy\right)⇒f_y(1,1)=1$. $f_{xx}=2-4y^2\pi ^2\cos\left(2\pi xy\right)⇒f_{xx}(1,1)=2-4\pi^2,$$f_{xy}=-2\pi \sin\left(2\pi xy\right)-4xy\pi ^2\,\cos\left(2\pi xy\right)⇒f_{xy}(1,1)=-4\pi^2,$$f_{yy}=-4x^2\pi ^2\cos\left(2\pi xy\right)⇒f_{yy}(1,1)=-4\pi^2$. Therefore $p_2(x,y)=3+2(x-1)+(y-1)+(1-2\pi^2)(x-1)^2-4\pi^2(x-1)(y-1)-2\pi^2(y-1)^2=(1-2\pi^2) x^2-4 \pi ^2 x y-2 \pi ^2 y^2+8 \pi ^2 x+(1+8 \pi ^2) y-8 \pi ^2+1$.