be2116…: “About a winter angle”

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Let $f:\R ^2 \to \R$ and let $z=f(x,y)$ be the surface associated to the function.

I. Classify the following as scalars, vectors in $\R ^2$ , or vectors in $\R ^3$ .

$f(3,5)$ is a scalarvector in $\R ^2$ vector in $\R ^3$ .
$\grad {f}(3,5)$ is a scalarvector in $\R ^2$ vector in $\R ^3$ .
$f_y(3,5)$ is a scalarvector in $\R ^2$ vector in $\R ^3$ .
The normal vector to the tangent plane to the surface at $(3,5,f(3,5))$ is a scalarvector in $\R ^2$ vector in $\R ^3$ .

II. Classify the following as subsets of $\R$ , $\R ^2$ , or $\R ^3$ .

The level curve of $f(x,y)$ through $(3,5)$ is a subset of $\R$ $\R ^2$ $\R ^3$ .
The domain of $f(x,y)$ is a subset of $\R$ $\R ^2$ $\R ^3$ .
The range of $f(x,y)$ is a subset of $\R$ $\R ^2$ $\R ^3$ .
The surface $z=f(x,y)$ is a subset of $\R$ $\R ^2$ $\R ^3$ .
The tangent plane to $z=f(x,y)$ at $(3,5,f(3,5))$ is a subset of $\R$ $\R ^2$ $\R ^3$ .