# Differential Geometry The gradient of a function $f$, $\tilde{d}f$ is a [[One-Form Field]], which when acting on an arbitrary [[Vector]] $d/d\lambda$ gives: $\tilde{d}f\left(\frac{d}{d\lambda}\right) = \frac{df}{d\lambda}$ More explicitly, the gradient of $f$ at a point $P \in M$, for some [[Manifold]] $M$, is the element of the [[Cotangent Space]] $T^*_P$ whose value on $\pmb{V} \in T_P$, where $T_P$ is the [[tangent space]], is the directional derivative of $f$ along a curve whose tangent is $\pmb{V}$. It is easy to see that the gradient is linear: $ \tilde{d}f\left(a \frac{d}{d\lambda} + b \frac{d}{d\mu}\right) = \left(a \frac{d}{d\lambda} + b \frac{d}{d\lambda}\right) f = a \frac{df}{d\lambda} + b \frac{df}{d\mu} = a \tilde{d}f(d/d\lambda) + b \tilde{d}f(d/d\mu) $ A pictorial representation of a topographical map is shown below, $\tilde{d}h$, where $h$ is the height. The gradient is largest in regions where the lines are close together (i.e. region $A$) and smallest in regions where they are further apart (i.e. region $B$). ![[Pasted image 20210124112844.png]]