# Geometric Definition

The definition of a tangent space is the space of all vectors tangent to a surface at some point.

# Tangent Spaces of Images

It turns out the span of the Jacobian is the tangent space to an image. In every direction $f_x$, the partial represents the instantaneous rate of change, so you can create a tangent line at that point using the partial. Therefore, in every direction, you have a tangent line. The tangent space is the span of all of these lines. $$\text{Span}\left(\begin{bmatrix}\frac{\partial f}{\partial x_1} & \frac{\partial f}{\partial x_2} & ... & \frac{\partial f}{\partial x_n} \end{bmatrix}\right)$$ Special case: $$f: \mathbb{R}^n \to \mathbb{R}$$ If this is the case, you can write $w = f(x,y,z) \to f(x,y,z) - w = 0$. In this case, it's the same type of problem as a level set.

# Example

Now, let's imagine a unit sphere: $$f(\theta, \phi) = < \cos(\theta)\sin(\phi), \sin(\theta)\sin(\phi), \cos(\phi)>$$ You can find how I got this here. Now, this is our Jacobian: $$\text{Span}\left(\begin{bmatrix}\frac{\partial f_1}{\partial \theta} & \frac{\partial f_1}{\partial \theta}\\ \frac{\partial f_2}{\partial \theta} & \frac{\partial f_2}{\partial \phi} \\ \frac{\partial f_3}{\partial \phi} & \frac{\partial f_3}{\partial \phi} \end{bmatrix}\right)$$ $$\text{Span}\left(\begin{bmatrix}-\sin(\theta)\sin(\phi) & \cos(\theta)\cos(\phi)\\ \cos(\theta)\sin(\phi) & \sin(\theta)\cos(\phi) \\ 0 & -\sin(\phi) \end{bmatrix}\right)$$ What answer do we expect? We expect to see a plane. The span of two vectors, by definition, is a plane.

# Tangent Spaces to Graphs

# Example

# Tangent Spaces to Level Curves

In the post about gradients, we talked about gradients were orthogonal to level curves. So, how do we combine our understanding of tangent spaces: the span of the derivative (Jacobian) and the span of the graph of the derivative.

# Example

David Witten