Tensor Calculus I

$\displaystyle \text{div} \ T=\lim_{V\to 0} \frac{1}{V}\oint T(d\vec{A})$

The divergence of a tensor field is defined as above for a vector field.T is a tensor viewed as a vector valued function of a vector variable,compute div T in rectangular coordinates using the $\hat{x},\hat{y}\ \text{and}\ \hat{z}$

1. $\partial T^{j\phi}/\partial x^{zr}$ in spherical coordinates

2. $\partial T^{ij}/\partial x^j$ in coordinates basis

3. $\partial T^{\mu\alpha}/\partial x^{zy}$ in coordinates basis where alpha $\alpha=i\pi\hat{z}$ extends to the complex plane

4. $\partial T^{\sum_{\gamma=0}^{\infty}n^{\gamma i}}/\partial x^j$ in rectangular coordinates

5.Not defined

6. $\infty -\sum_{n=V} \frac{\partial T}{\partial V}$ in coordinate basis

7. $\partial V^{j}/\partial T^j$

8. $\text{div} V - \text{div} T$

Note :It requires the addition of vectors at two different points,so it must be assumed you can move a vector over parallel to itself and add it to a vector at another point.