Chapter 1.6, Problem 1.61P

(a)

To determine

To show: ${\displaystyle \underset{V}{\int }\left(\nabla \cdot T\right)d\tau }={\displaystyle \underset{S}{\oint }T\cdot da}$

(b)

To determine

To show: ${\displaystyle \underset{V}{\int }\left(\overrightarrow{\nabla }\times \overrightarrow{v}\right)d\tau }=-{\displaystyle \underset{S}{\oint }\overrightarrow{v}\times da}$

(c)

To determine

To show: ${\displaystyle \underset{V}{\int }\left[T{\nabla }^{2}U+\left(\nabla \cdot T\right)\cdot \left(\nabla \cdot U\right)\right]}d\tau ={\displaystyle \underset{S}{\oint }\left(T\nabla U\right)\cdot da}$

(d)

To determine

To show: ${\displaystyle \underset{V}{\int }\left[T{\nabla }^{2}U-U{\nabla }^{2}T\right]}d\tau ={\displaystyle \underset{S}{\oint }\left(T\nabla \cdot U-U\nabla \cdot T\right)\cdot da}$

(e)

To determine

To show: ${\displaystyle \underset{S}{\int }\overrightarrow{\nabla }\cdot \overrightarrow{T}\times da}={\displaystyle \underset{P}{\oint }\overrightarrow{T}\cdot dl}$