Functions of several variables

Determine and describe the domain of

$$f(x,y)=\sqrt{9-x^2-y^2}.$$

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Determine and describe the domain of

$$f(x,y)=\ln (x+y-1).$$

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Determine and describe the domain of

$$f(x,y)=\frac{\sqrt{y-x^2}}{x^2+y^2-4}.$$

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Determine the level curves of

$$f(x,y)=x^2+y^2+2y.$$

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Determine the level curves of

$$f(x,y)=e^{xy}.$$

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Determine whether the following limit exists

$$\underset{(x,y)\to (0,0)}{\lim }\frac{3x^2-y^2}{x^2+y^2}.$$

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Determine whether the following limit exists

$$\underset{(x,y)\to (0,0)}{\lim }\frac{x^{3}y}{x^6+y^2}.$$

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Compute the limit

$$\underset{(x,y)\to (0,0)}{\lim }\frac{x{y}^2}{x^2+y^2}.$$

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Compute the limit

$$\underset{(x,y)\to (0,0)}{\lim }\frac{x^4+y^4}{x^2+y^2}.$$

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Compute the limit

$$\underset{(x,y)\to (0,0)}{\lim }\frac{e^{x^2+y^2}-1}{x^2+y^2}.$$

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Study the continuity on ${\mathbb{R}}^2$ of the function

$$f(x,y)=\{\begin{array}{ll}\frac{y^3}{x^2+y^2}& (x,y)\ne (0,0)\\ 0& (x,y)=(0,0)\end{array}$$

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Study the continuity on ${\mathbb{R}}^2$ of the function

$$f(x,y)=\{\begin{array}{ll}\frac{(x+y{)}^2}{x^2+y^2}& (x,y)\ne (0,0)\\ 1& (x,y)=(0,0)\end{array}$$

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Determine whether the function

$$f(x,y)=\frac{\sin \left(3(x^2+y^2)\right)}{x^2+y^2},(x,y)\ne (0,0),$$

can be extended continuously to the origin.

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Determine whether the function

$$f(x,y)=x^2-xy+y^2$$

attains an absolute maximum and minimum on the set $K=\{(x,y)\in {\mathbb{R}}^2:x^2+y^2\le 1\}$.

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