Vault Of Euclid

Determining if a Surface Parameterization is Smooth

Multivariable calculus core

Is the surface parameterization \(\mathbf{r}(u, v) = \langle u^{2v}, v+1, \sin u \rangle, 0 \leq u \leq 2, 0 \leq v \leq 3\) smooth?

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What you need first

Partial derivatives
Cross product of vectors
Derivatives of exponential functions with arbitrary bases

What this problem tests

Tests the ability to compute partial derivatives of vector-valued functions, calculate cross products, and apply the definition of a smooth surface.

This problem is typical in a Multivariable Calculus course when introducing parameterized surfaces and surface integrals.

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