**Question:** "Sorry but I have a couple of questions: Why is c(1)c(2)=c2(V) and why is c1(V)=c(1)+c(2)? And also, when we write c using these t's, they sort of serve to indicate what term corresponds to which ci right? @hm2020"

**Answer:** when $r=2$ the formula follows from your relation $c(V \otimes L)=\prod_j (1 + c_1(L_j') + c_1(L))$: Let $c(i):=c_1(L_i)$ and $c:=c_1(L)$. You get the calculation

$$c_t(V\otimes L)=(1+(c(1)+c)t)(1+(c(2)+c)t)= $$

$$...+(c(1)c(2)+(c(1)+c(2))c+c^2)t^2=$$

$$\cdots + (c_2(V)+c_1(V)c_1(L)+\binom{2}{2}c_1(L)^2)t^2=$$

$$c_0(V\otimes L) +c_1(V\otimes L)t+c_2(V\otimes L)t^2.$$

**Note:** You get

$$c_t(V)=(1+c(1)t)(1+c(2)t)= $$

$$1+(c(1)+c(2))t+c(1)c(2)t^2=c_0(V)+c_1(V)t+c_2(V)t^2$$

hence

$$c_1(V)=c_1(L_1)+c_1(L_2)\text{ and }c_2(V)=c_1(L_1)c_1(L_2).$$

Here you view the elements as "living" in a commutative ring and multiply (they all live in the even cohomology ring $H^{2*}(X)$ which is commutative) You find this explained in Hartshorne, Appendix A.

**Note:** Formulas for Chern classes
$c_i(E\otimes F)$ where $E,F$ have "Chern roots" $a_i,b_j$ are expressed in terms of polynomials in the roots $a_i,b_j$. The coefficient of the $t^i$ term "lives" in the group $H^{2i}(X)$. Whenever you have a theory of Chern classes $c_i(E) \in H^{2i}(X)$ living in the even part of a cohomology ring $H^*(X)$ you may perform such calculations. The experession

$$c_t(E):=c_0(E)+c_1(E)t+\cdots + c_r(E)t^r$$

"lives" in

$$H^0(X)\oplus H^2(X)t\oplus \cdots \oplus H^{2r}(X)t^r.$$

For any rank $r$ locally trivial sheaf $E$ there is the complete flag bundle $F(E)$ of $E$ and an injection

$$H^*(X) \subseteq H^*(F(E)),$$

hence you may perform all calculations in $H^*(F(E))$. In $H^*(F(E))$
you have the equality

$$c_r(E)=c_r(L_1)+ \cdots + c_r(L_r)$$

where $L_i$ are invertible sheaves. You get the formula

$$c_t(E)=c_t(L_1)\cdots c_t(L_r)$$

in $H^{2*}(X)[t]$: It holds in $H^{2*}(F(E))[t]$ but since $c_t(E) \in H^{2*}(X)[t]$ the equality holds here.