Q. 35 - Calculus

Question

Using Theorem as a model, provide a conjecture you think would be sufficient to guarantee that a function of variables is differentiable at a point in its domain.

Using Theorem $12.31$as a model, provide a conjecture you think would be sufficient to guarantee that a function of $n$ variables is differentiable at a point in its domain.

Step-by-Step Solution

Answer

There is no assurance that the incremental application of Martin's Theorem in the tangent plane will be correct or even well defined if the surface for a desired purposes of Reynolds' Theorem is not smooth or piecewise smooth. As a result, the presentation of Stokes' Theorem necessitates the smoothness or piecewise smoothness of the surface$S$

Step: 1 SToke theorm: (part a)

The goal is to figure out why the formulation of Stokes' Theorem demands a smooth or piecewise smooth surface $S$ What is wrong if this criterion is not met?

"Let$S$ be an oriented, smooth or piecewise-smooth surface bordered by a curve C," says Stokes. Assume that $mathbfn$ is an orientated unit normal vector of $S$, and that $C$ has a parametrization that traverses $C$ counterclockwise in relation to $mathbfn$.

Step: 2 stoke's (part a)

The goal is to figure out why the formulation of Stokes' Theorem demands a smooth or piecewise smooth surface $S$. What is wrong if this criterion is not met?

"Let $S$ be an oriented, smooth or piecewise-smooth surface bordered by a curve C," says Stokes. Assume that $mathbfn$ is an orientated unit normal vector of $S$, and that $C$ has a parametrization that traverses $$C$$ counterclockwise in relation to $mathbfn$.

Step: 3

The goal is to figure out why the formulation of Stokes' Theorem demands a smooth or piecewise smooth surface $S$. What is wrong if this criterion is not met?

"Let $S$ be an oriented, smooth or piecewise-smooth surface bordered by a curve C," says Stokes. Assume that $mathbfn$ is an orientated unit normal vector of $S$, and that $C$ has a parametrization that traverses $C$ counterclockwise in relation to $mathbfn$.