Chart independence and the transformation law of a smooth covariant 2-tensor field

Let $p\in U\cap V$ and put $y_0:=y(p)$. For every smooth function $f$ defined near $p$ the coordinate derivation is $(\mathrm{\partial }\mathrm{/}\mathrm{\partial }{y}^a{\mathrm{\mid }}_p)f=D_a(f\circ y^{-1})(y_0)$ by [L1], and on a neighbourhood of $y_0$ the representative factors as $f\circ y^{-1}=(f\circ x^{-1})\circ (x\circ y^{-1})$ with $x\circ y^{-1}$ smooth by [L3]. The chain rule [L2] applied at $y_0$ gives $D_a(f\circ y^{-1})(y_0)={\sum }_c{D}_c(f\circ x^{-1})(x(p))D_a(x\circ y^{-1}{)}^c(y_0)$, that is, $\frac{\mathrm{\partial }}{\mathrm{\partial }{y}^a}{\mid }_{p}f={\sum }_c\frac{\mathrm{\partial }{x}^c}{\mathrm{\partial }{y}^a}(y_0)\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^c}{\mid }_{p}f$, where $\frac{\mathrm{\partial }{x}^c}{\mathrm{\partial }{y}^a}(y_0):=D_a(x\circ y^{-1}{)}^c(y_0)$.

Both sides of the identity in step 1.1 act on smooth germs at $p$ and are derivations there, so the coordinate vector fields are related by $\frac{\mathrm{\partial }}{\mathrm{\partial }{y}^a}{\mid }_p={\sum }_c\frac{\mathrm{\partial }{x}^c}{\mathrm{\partial }{y}^a}(y_0)\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^c}{\mid }_p$. Evaluating the bilinear form $A_p$ on the pair $(\frac{\mathrm{\partial }}{\mathrm{\partial }{y}^a}{\mid }_p,\frac{\mathrm{\partial }}{\mathrm{\partial }{y}^b}{\mid }_p)$ and using bilinearity together with [L5] gives $A_{ab}^{\mathrm{\prime }}(y_0)={\sum }_{c,d}\frac{\mathrm{\partial }{x}^c}{\mathrm{\partial }{y}^a}(y_0)\frac{\mathrm{\partial }{x}^d}{\mathrm{\partial }{y}^b}(y_0)A_{cd}(x(p))$, which is the displayed transformation law at $p$.

Fix an arbitrary chart $(V,y)$ of the maximal atlas and a point $p\in V$. Since the supplied smooth atlas covers $M$, there is a chart $(U_0,x_0)$ in that atlas with $p\in U_0$. On the open overlap $U_0\cap V$, the transition $x_0\circ y^{-1}$ is smooth by [L3], so its first partial derivatives are smooth; each component function $A_{cd}^{x_0}=A_{cd}\circ x_0^{-1}$ is smooth on $x_0(U_0)$ by the hypothesis and [L6]. The transformation law of step 2.1 expresses each $A_{ab}^y$ on $y(U_0\cap V)$ as a finite sum of products of these derivative functions and $A_{cd}^{x_0}\circ (x_0\circ y^{-1})$. These products and compositions are smooth by [L4]. Thus the components are smooth on a neighbourhood of $y(p)$. Since $p$ was arbitrary, they are smooth throughout $y(V)$.

The maximal-atlas chart $(V,y)$ in step 3.1 was arbitrary, so the components of $A$ are $C^{\mathrm{\infty }}$ in every chart of the maximal atlas whenever they are $C^{\mathrm{\infty }}$ in the charts of one atlas. Step 2.1 proves the transformation law for every pair of overlapping charts. Thus $A$ is a smooth covariant $2$-tensor field by [L5], as claimed.