Suppose we have the paraboloid $z=x^2+y^2$ and the plane $z=y$. Their intersection produces a curve $C$, and certain surfaces bounded by it, for example the disc $S$ which directly fills the area of $C$ and the paraboloid $S’$ given by $z=x^2+y^2$ which extends from $C$ downwards and is bounded by $C$. My question is on […]

Find the parametric form $S(u, v)$ where $a \le u \le b$ and $c \le v \le d$ for the triangle with vertices $(1, 1, 1), (4, 2, 1),$ and $(1, 2, 2)$. I am told that one parameterisation is $S(u, v) = (1 + 3u, 1 + u + v, 1 + v)$ for […]

How can I prove the following? $\gamma (t)$ is unit speed, $\dot \gamma(t) \not= 0 \Rightarrow \ddot \gamma(t)$ is perpendicular to $\gamma(t)$ I don’t really see where a problem would arise when $\dot \gamma(t)=0$ would cause such a problem

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