Difference between tangent space and tangent plane

Question

I’ve avoided doing any manifold (regretting it somewhat) courses, however do have some understanding. Let $p$ be a point on a surface $S:U\to \Bbb{R}^3$, we define:

The tangent space to $S$ at $p$, $T_p(S)=\{k\in\Bbb{R}^3\mid\exists\textrm{ a curve }\gamma:(-ε,ε)\to S\textrm{ with }\gamma(0)=p,\gamma'(0)=k\}$.

The tangent plane to $S$ at $p$ as the plane $p+T_p(S)\subseteq\Bbb{R}^3$.

My current understanding is, in the diagram below the tangent plane is the plane shown, whilst the tangent space would be p minus each element of the plane, hence the corresponding plane passing through the origin. Is this correct or is it incorrect? I’m doing a course called geometry of curves and surfaces and being unsure about this is making understanding later topics difficult.

Edit - can't post images, here's a link instead! http://standards.sedris.org/18026/text/ISOIEC_18026E_SRF/image022.jpg

Thanks!

Answer 1

While the definitions you've given are acceptable, I would use different definitions for tangent space and tangent plane that reveal some more mathematical structure.

• The tangent plane is a geometric object. You can define the tangent plane to a point $p$ on a surface $S$ in $\mathbb{R}^3$ as a plane in $\mathbb{R}^3$ which intersects $p$ and whose normal vector is parallel to the gradient of $f$, if $S$ is represented as the level surface $f(x,y,z) = 0$. But all that formality is simply used to the describe the geometric object that everybody can visualize intuitively.

• The tangent space is a vector space of linear functionals. Each vector $v \in T_p S$ acts on a function $g \in C^1(S)$ in a way such that $v(g)$ gives you a directional derivative of $g$ in a particular direction (which can be identified with $v$ itself). Thus the tangent space $T_p S$ can be naturally identified with the tangent plane, because the only directions that make sense to take the directional derivative on are the directions that lie on the tangent plane.

Answer 2

Exactly. The tangent space is simply a translation of the tangent plane by taking $p$ to $0$. One nice property that the tangent space always has (but which a tangent plane almost never does) is that it is closed under addition and scalar multiplication--that is, for any $\alpha\in\Bbb R$ and any $x,y$ in the tangent space, we have $x+y$ and $\alpha x$ in the tangent space.