- #1

docnet

Gold Member

- 799

- 485

- Homework Statement
- I need to compute this partial derivative

- Relevant Equations
- ##\partial_tu\big[(t,x-t\kappa V)\big]##

Hi all, I was wondering is if the following partial derivative can be computed without a specific ##u(t,x)##

$$\partial_tu\big[(t,x-t\kappa V)\big]$$

I was thinking it can't be done, because we could have

$$u_a(t,x)=tx \Rightarrow \partial_tu\big[(t,x-t\kappa V)\big]=\partial_t\big[tx-t^2\kappa V\big]=x-2t\kappa V$$

$$u_b(t,x)=t+x \Rightarrow \partial_tu\big[(t,x-t\kappa V)\big]=\partial_t\big[t+x-t\kappa V\big]=1-\kappa V$$

so there is no universal formula for ##\partial_tu\big[(t,x-t\kappa V)\big]##, which depends on the function ##u(t,x)##.

$$\partial_tu\big[(t,x-t\kappa V)\big]$$

I was thinking it can't be done, because we could have

$$u_a(t,x)=tx \Rightarrow \partial_tu\big[(t,x-t\kappa V)\big]=\partial_t\big[tx-t^2\kappa V\big]=x-2t\kappa V$$

$$u_b(t,x)=t+x \Rightarrow \partial_tu\big[(t,x-t\kappa V)\big]=\partial_t\big[t+x-t\kappa V\big]=1-\kappa V$$

so there is no universal formula for ##\partial_tu\big[(t,x-t\kappa V)\big]##, which depends on the function ##u(t,x)##.