Q. 35 - Calculus

Question

The objective is to find why the statement of Stokes' Theorem requires that the surface \(S\) is smooth or piecewise smooth. If this condition is not met, what wrong.

Stokes' Theorem states that,

"Let \(S\) be an oriented, smooth or piecewise-smooth surface bounded by a curve C. Suppose that \(\mathbf{n}\) is an oriented unit normal vector of \(S\) and \(C\) has a parametrization that traverses \(C\) in the counterclockwise direction with respect to \(\mathbf{n}\).

Step-by-Step Solution

Answer

The objective is to find why the statement of Stokes' Theorem requires that the surface $S$ is smooth or piecewise smooth. If this condition is not met, what wrong.

Stokes' Theorem states that,

"Let $S$ be an oriented, smooth or piecewise-smooth surface bounded by a curve C. Suppose that $\mathbf{n}$ is an oriented unit normal vector of $S$ and $C$ has a parametrization that traverses $C$ in the counterclockwise direction with respect to $\mathbf{n}$.

Step: 1

Calculate the exact value of each definite integral in Exercises 47-52 by using properties of definite integrals and the formulas in Theorem $4.13$.

${\int }_C \mathbf{F}(x,y,z)\times d\mathbf{r}=\frac{5}{12}$

$={\int }_0^1 \left(-3x^5+3x^3-x^2+x\right)dx$

$=\left(-\frac{1}{2}\times x^6+\frac{3}{4}\times x^4-\frac{x^3}{3}+\frac{x^2}{2}\right)-\left(-\frac{1}{2}\times x^6+\frac{3}{4}\times x^4-\frac{x^3}{3}+\frac{x^2}{2}\right)$

Step: 2

1234

Step: 3

124