Section 11.2 Problem 10: Find the limit, if it exists, or show that the limit does not exist. \lim_{(x,y)\to(0,0)}\frac{x^2\sin^2y}{x^2+2y^2}.
Comment: Notice that for 0\leq\frac{x^2\sin^2y}{x^2+2y^2}=\left(\frac{x^2}{x^2+2y^2}\right)\sin^2y. Then apply the squeeze theorem.
Section 11.2 Problem 26: Determine the set of points at which the function is continuous. f(x,y,z)=\sqrt{y-x^2}\ln z.
Comment: As the function is a composition of several continuous function, it is continuous on its domain.
Section 11.2 Problem 28: Determine the set of points at which the function is continuous. f(x,y)=\begin{cases}\frac{xy}{x^2+xy+y^2}&\text{if }(x,y)\neq(0,0)\\0&\text{if }(x,y)=(0,0)\end{cases}.
Comment: Apparently f is continuous on \mathbb{R}^2-\{(0,0)\}. To see if it is continuous at (0,0), we need to check \lim_{(x,y)\to(0,0)}f(x,y)=0.
Section 11.2 Problem 30: Use polar coordinates to find the limit. \lim_{(x,y)\to(0,0)}(x^2+y^2)\ln(x^2+y^2).
Comment: Let r=\sqrt{x^2+y^2}. Then \lim_{(x,y)\to(0,0)}(x^2+y^2)\ln(x^2+y^2)=\lim_{r\to 0}r^2\ln(r^2). Then apply the l’Hospital’s rule.
Section 11.3 Problem 26: Find the first partial derivatives of the function. u=x^{y/z}.
Comment: Recall that (x^a)'=ax^{a-1}.
Section 11.3 Problem 55: Find the indicated partial derivative. f(x, y, z)=e^{xyz^2}; f_{xyz}.
Solution: First f_x=yz^2f. Second f_{xy}=z^2f+yz^2f_y=z^2f+yz^2xz^2f=(z^2+xyz^4)f. Last f_{x,y,z}=(2z+4xyz^3)f+(z^2+xyz^4)f_z=(2z+4xyz^3)f+(z^2+xyz^4)2xyzf=2z(1+3xyz^2+x^2y^2z^4)e^{xyz^2}.
Section 11.3 Problem 64a: Show that each of the following functions is a solution of the wave equation u_{t t} = a^2u_{xx}. u=\sin(kx)\sin(akt).
Comment: Use the expression to get u_{tt}, u_{xx}. Check the equation holds.