Proof of distributive and product rule

[srhelfrich]

1. Prove a) r=(u*v)=r*u+r*v and b) d/dt(r*s)=r*ds/st+dr/dt*s
2. Homework Equations : b) dr/dt=lim t->0=Δr/Δt and Δr=r(t+Δt)-r(t)
3. Attempt at the solution:
Okay, so I was able to work out part a but I'm not quite sure how to start part b. Could anyone point me toward a useful resource to explain how to approach this problem? I'm not looking for an answer, just a means to an answer.

(* is dot product)

 

[gopher_p]

1. Prove a) r=(u*v)=r*u+r*v and b) d/dt(r*s)=r*ds/st+dr/dt*s



2. Homework Equations : b) dr/dt=lim t->0=Δr/Δt and Δr=r(t+Δt)-r(t)



3. Attempt at the solution:
Okay, so I was able to work out part a but I'm not quite sure how to start part b. Could anyone point me toward a useful resource to explain how to approach this problem? I'm not looking for an answer, just a means to an answer.

(* is dot product)

Are you familiar with a proof of the standard product rule; i.e. the version that says (fg)'=f'g+fg'? The proof of this version shouldn't look significantly different.

 

[srhelfrich]

Thank you! I looked it up and you're right, I see where they start with the limit and expand from there. Much appreciated!

 

[gopher_p]

Alright then. The key things you'll want to verify (if you haven't already) are that the algebra $$\mathbf{u}\cdot(\mathbf{v}+\mathbf{w})=\mathbf u\cdot\mathbf v+\mathbf{u}\cdot\mathbf{w},\ \ c(\mathbf{u}\cdot\mathbf{v})=(c\mathbf{u})\cdot\mathbf{v}$$ and calculus
$$\lim_{t\rightarrow a}\left(\mathbf u(t)+\mathbf v(t)\right)=\lim_{t\rightarrow a}\mathbf u(t)+\lim_{t\rightarrow a}\mathbf v(t), \ \ \lim_{t\rightarrow a}\left(\mathbf u(t)\cdot\mathbf v(t)\right)=\lim_{t\rightarrow a}\mathbf u(t)\cdot\lim_{t\rightarrow a}\mathbf v(t)$$
work the same with vectors/vector-valued functions as they do with numbers/real-valued functions.

 

[HallsofIvy]

1. Prove a) r=(u*v)=r*u+r*v

This certainly isn't true! You mean "r*(u+ v)= r*u+ r*v" don't you?

and b) d/dt(r*s)=r*ds/st+dr/dt*s

2. Homework Equations : b) dr/dt=lim t->0=Δr/Δt and Δr=r(t+Δt)-r(t)



3. Attempt at the solution:
Okay, so I was able to work out part a but I'm not quite sure how to start part b. Could anyone point me toward a useful resource to explain how to approach this problem? I'm not looking for an answer, just a means to an answer.

(* is dot product)