I need to solve this problem for my vector calculus class.
Suppose $\gamma :[a,b]\rightarrow \mathbb{R}^3$ is $C^1$ path of length $L$ and $\mathbf{F}$ is a $C^1$ vector field in $\mathbb{R}^n$ with $\| \mathbf{F}(x,y,z)\| \leq M,$ $\forall(x,y,z)\in \mathbb{R}^3$. Prove that $$|\int_{\gamma}\mathbf{F}\cdot \,ds | \leq LM$$
I can see why this is true. Indeed, the integral represents the work done by the vector field to make a particle "move" along $\gamma$. It is therefore logical that by multiplying the value of this vector field to the length of the path we obtain the integral in question. However, I have no idea how to prove it. I tried to see if Green's theorem, Stokes' theorem or Gauss theorem could help me but, if so, I don't see how.
Thanks in advance for any help you can give me.