## Derivations

Let $𝔽$ be a field. A vector space over $𝔽$ is an abelian group $V$ with a function $𝔽×V ⟶ V (c,v) ⟼ cv$ such that

1. if ${c}_{1},{c}_{2}\in 𝔽$ and $v\in V$ then $\left({c}_{1}+{c}_{2}\right)v={c}_{1}v+{c}_{2}v$,
2. if $c\in 𝔽$ and ${v}_{1},{v}_{2}\in V$ then $c\left({v}_{1}+{v}_{2}\right)=c{v}_{1}+c{v}_{2}$,
3. if ${c}_{1},{c}_{2}\in 𝔽$ and $v\in V$ then ${c}_{1}\left({c}_{2}v\right)=\left({c}_{1}{c}_{2}\right)v$, and
4. if $v\in V$ then $1\cdot v=v$.

Let $𝔽$ be a field. Let $V,W$ be vector spaces over $𝔽$. An $𝔽$-linear map from $V$ to $W$ is a function $\phi :V\to W$ such that

1. $\phi$ is a group homomorphism,
2. if $c\in 𝔽$ and $v\in V$ then $\phi \left(cv\right)=c\phi \left(v\right)$,

Let $𝔽$ be a field. An algebra is a vector space $A$ over $𝔽$ with an function $A×A ⟶ A (a1,a2) ⟼ a1a2$ such that $A$ is a ring and scalar multiplication is the composition of the map $𝔽 ⟶ A ξ ⟼ ξ⋅1$ and the multiplication in $A$.

Let $𝔽$ be a field. Let $A$ be an $𝔽$-algebra. A derivation of $A$ is an $𝔽$-linear map $d:A\to A$ such that $if a1,a2 ∈A then d(a1a2) = a1d(a2) +d(a1)a2 .$

1. There is a unique derivation $\frac{d}{dx}$ if $𝔽\left[x\right]$ such that $\frac{dx}{dx}$ =1.
2. If $p\in 𝔽\left[x\right]$ then $dpdx = (coefficient of y in p(x+y) ).$
3. If $p\in 𝔽\left[x\right]$ then $p= ∑k∈ ℤ≥0 ( (ddx) k p ) | x=0 xk$
4. There is a unique extension of $\frac{d}{dx}$ to a derivation of $𝔽\left(x\right)$.
5. There is a unique extension of $\frac{d}{dx}$ to a derivation of $𝔽\left[\left[x\right]\right]$.
6. There is a unique extension of $\frac{d}{dx}$ to a derivation of $𝔽\left(\left(x\right)\right)$.
7. If $p\in 𝔽\left[\left[x\right]\right]$ then $dpdx = (coefficient of y in p(x+y) ).$
8. If $p\in 𝔽\left[\left[x\right]\right]$ then $p= ∑k∈ ℤ≥0 ( (ddx) k p ) | x=0 xk$
9. ## Notes and References

These notes provide a bridge between an introductory calculus course and the use of derivations in the definitions of tangent spaces in algebraic geometry.

## References

[BouTop] N. Bourbaki, General Topology, Chapter VI, Springer-Verlag, Berlin 1989. MR?????