Frankel 2.4 Problems

(1)

The second rank tensor given by $$a_ib_j dx^i \otimes dx^j$$ has the value on vectors $$v$$ and $$w$$:

$$a_ib_j dx^i(v) \otimes dx^j(w) = a_iv^i b_jw^j.$$

For one-forms $$\alpha$$ and $$\beta$$ with components $$a_i$$ and $$b_j$$, the tensor product $$\alpha \otimes \beta$$ has the value on vectors $$v$$ and $$w$$:

$$\alpha(v) \otimes \beta(w) = a_iv^i b_jw^j.$$

Thus,

$$\alpha \otimes \beta = a_ib_j dx^i \otimes dx^j.$$

(2)

let $$\mathbf{A} : E \to E$$ be a linear transformation.

(i) The trace tr($$\mathbf{A}) = A_i^i$$. If we change coordinate systems from $$x$$ to $$y$$ the components of the mixed tensor of $$\mathbf{A}$$ transform as

$$A'^l_k = \frac{\partial x^j}{\partial y^k}\frac{\partial y^l}{\partial x^i}A^i_j.$$

Therefore, tr($$\mathbf{A})$$ is a scalar:

$$A'^k_k = \frac{\partial x^i}{\partial y^k}\frac{\partial y^k}{\partial x^i}A^i_i = A^i_i,$$

(3)

Let $$\vec{v} = v^i\partial_i$$ be a vector on $$M^n$$.

(i) We define a new object by $$v_j = g_{ji}v^i$$. By the transformation properties,

$$v'_k = \frac{\partial x^i}{\partial y^k}\frac{\partial x^j}{\partial y^k}g_{ji}\frac{\partial y^k}{\partial x_i}v^i = \frac{\partial x^j}{\partial y^k}g_{ji}v^i = \frac{\partial x^j}{\partial y^k}v_j,$$

thus, $$v_j$$ defines a one-form.