# Formula of curvature not defined in arc length

As a title I have some trouble to understand the formula of curvature that professor gave to us during the lesson. I have already tried to read the book he suggested and searched on the internet but I did not find what I was looking for...

Firstable he gave us the definition of curvature using the arc lenght.

Given the curve $$\gamma(t): [a,b] \to \mathbb{R}$$, the curvature vector $$\vec k(s)$$ is defined as $$\vec k(s)=\frac{d}{ds}\tau(s)=\frac{d^2}{ds^2}\gamma(s)$$

Later, since usually arc lenght parametrization is less commonly used, he wrote down this:

Thank to the chain rule and the definition of function $$s(t):[a,b] \to [0,L(\gamma)]$$ in the arc lenght parametrization we obtain: $$\partial_s = \frac{1}{|\gamma'(t)|}\partial_t$$ Now we using the definition of curvature vector $$\vec k(s)$$: $$\partial_s^2 = \vec k = \frac{1}{|\gamma'(t)|}\partial_t \gamma_s = \frac{1}{|\gamma'(t)|} \left( \frac{\gamma'(t)}{|\gamma'(t)|} \right)' = \frac{\gamma''(t)- \langle \gamma''(t),\tau(t) \rangle \cdot \tau(t)}{|\gamma'(t)|^2}$$

I do not understand the last equality, esentially for two reasons.

• The math process that the professor has followed to write down this result (I do not know how to do a derivate of a quotient in more than one variable)
• I know that the derivate of a vector is another vector orthogonal to the first. For this reason I think that the scalar product should be zero.

In the end he said that $$P_{\gamma(x)}^{\perp}\left( \frac{\gamma''(t)}{|\gamma'(t)|^2} \right) = \frac{\gamma''(t)- \langle \gamma''(t),\tau(t) \rangle \cdot \tau(t)}{|\gamma'(t)|^2}$$ By definition: $$P_{\gamma(x)}^{\perp}(\eta) := \eta - \langle \eta, \tau \rangle \space \tau$$ where $$\eta$$ is a casual vector in $$\mathbb{R^n}$$ that has its application point on the curve; $$\tau$$ is the tangent vector at the point curve $$x$$. As the professor said $$P_{\gamma(x)}^{\perp}(\eta)$$ is the projection of vector $$\eta$$ to the normal vector of the curve point $$x$$, but I can not see why.

However I'm not pretty sure about the last definition and I can't find it nowhere (because I do not even know its name). Can somebody teach me what $$P_{\gamma(x)}^{\perp}(\eta)$$ is, please?