Find dF/ds in Terms of u, v & w - Conditions and Assumptions

[gomunkul51]

Hi!

As a part of a mathematical construction I used the rate of change of F(s) with s, i.e. dF/ds. F and s are smooth functions. The problem is that s(u,v,w) is a function of u, v & w and actually I need to find the rate of change of F with respect to u, v & w. The question is: what does dF/ds equals to in terms of u, v & w ?

Thank you in advance !

** Please also provide the conditions and assumptions for the transition.

 

[lavinia]

Hi!

As a part of a mathematical construction I used the rate of change of F(s) with s, i.e. dF/ds. F and s are smooth functions. The problem is that s(u,v,w) is a function of u, v & w and actually I need to find the rate of change of F with respect to u, v & w. The question is: what does dF/ds equals to in terms of u, v & w ?

Thank you in advance !

** Please also provide the conditions and assumptions for the transition.

In this case F must be differentiated in a direction. Different directions will give different derivatives. All of the directional derivatives can be expressed in terms of derivatives in the directions of u,v, and w. This remarkable fact makes multivariate calculus possible.

 

[gomunkul51]

In this case F must be differentiated in a direction. Different directions will give different derivatives. All of the directional derivatives can be expressed in terms of derivatives in the directions of u,v, and w. This remarkable fact makes multivariate calculus possible.

F is smooth, i.e. differentiable any number of times in any direction for u,v, and w.
s is a known smooth function of u,v, and w. I need dF/ds in terms that include u,v, and w.

Can you tell me then how do I construct it?

It is probably related to dF/du + dF/dv + dF/dw but not equal to it.

 

[lavinia]

F is smooth, i.e. differentiable any number of times in any direction for u,v, and w.
s is a known smooth function of u,v, and w. I need dF/ds in terms that include u,v, and w.

Can you tell me then how do I construct it?

It is probably related to dF/du + dF/dv + dF/dw but not equal to it.

I think u,v,and w are functions of the parameter,s,