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# Vector calculus: test 1

These are exercises for Vector calculus: course.

1. Find and classify the critical points of the function $% f(x,y)=x^{4}+y^{4}-4xy+1.$

2. Locate, using Lagrange multipliers, the points where the function $% f(x,y)=x^{2}+2y^{2}$ attains its global maximum and minimum values on the circle $x^{2}+y^{2}=1.$

3. Find the best affine approximation of the vector function $% F(x,y)=(xy,x^{2}+y^{2})$ at the point where $x=1$ and $y=1.$

4. Find the volume of the solid in $\mathbf{R}^{3}$ bounded by the following surfaces: $z=x^{2}+y^{2},$ $z=0,$ $y=x^{2},$ $y=2x.$

5. By means of an appropriate change of variables, evaluate the integral $% \int_{D}xydA,$ where $D\subset \mathbf{R}^{2}$ is bounded by the lines $x=0,$ $x=1,$ $y=x,$ $y=x+1.$

6. Do (a) or (b).

(a) Using the $\varepsilon -\delta $ definition, prove that the composition of two continuous functions $f:\mathbf{R}^{n}\rightarrow \mathbf{R}^{m}$ and $g:\mathbf{R}^{m}\rightarrow \mathbf{R}^{p}$ is continuous.

(b) Suppose $f,g:B\rightarrow \mathbf{R}$ are continuous functions, where $B$ is a disk in $\mathbf{R}^{2},$ such that \begin{equation*} \begin{array}{l} \text{(1) }f(x)\geq g(x)\text{ for all }x\in B, \\ \text{(2) }\int_{B}f(x)dA=\int_{B}g(x)dA.% \end{array} \end{equation*} Using the properties of the integral and continuity, show that $f(x)=g(x)$ for all $x\in B.$