What if it's not a unit vector in directional derivatives

[kochibacha]

i came to this topic and they said

D_uf(x) = ||gradient vector|| * ||U|| * cos 0

if ||U|| were not a unit vector it would give different rate of change of f in any direction

what would happens if used ||U|| = 10 ?

 

[HallsofIvy]

Are you surprised? $\nabla f(x)*cos(\theta)$, where $$\theta$$ is the direction of vector u, is defined as the rate of change of vector function f(x) in the direction of vector v. $D_uf(x)$ is the rate of change of vector function f "with respect to vector u" which is essentially using u as the unit of measurement in the same way we define "df/du", with f and u scalar functions, using the "chain rule" in Calculus I. To go from "in the direction" to "with respect to the vector" we have to include the length of the vector, ||u||, as well.

 

[kochibacha]

could you help me discuss with this problem , considering a real physical problem would get me a better understanding of what does this mean

from Ideal gas law

PV=nRT

where P = pressure , V= volume , T = temperature, n= constant=1 , R = constant =1
let n = R = 1 , we have P=T/V

what rate should the temperature and volume be changing to make the rate of change of pressure fastest at point T = 25 , V = 10 ?

differentiate with respect to t

dP/dt = ∂P/∂T * dT/dt + ∂P/∂V * dV/dt......∂P/∂T = 1/V , ∂P/∂V = -T/V²


if i set T = 25 +t , V = 10+2t , i.e tempertature starting at 25 degree celsius with increment of 1 degree per minute and volume of 10 cm³ with increment of 2 cm³ per minute

from this point ∇f(x) would be < 1/V , -T/V²> = < 1/10 ,-25/100>

and U^→ = < dT/dt, dV/dt > = < 1,2 >

Ok now, if i substitute all above and calculate dP/dt = D_(1,2)(25,10) = 1/10-1/2 = -4/5 so the pressure is decreasing at rate of -4/5 unit when volume is raised 2 times faster than temperature at the point V= 10 cm³ , T= 25 degree

∇f(x)= < 1/10 ,-25/100> is the direction where rate of change of pressure is fastest which is

√(1/100+1/16) = 0.27 (i don't even know if it's decreasing or increasing at this rate) and it's less than 4/5 if i scale down <1 , 2> to unit vector it will be -0.18 but V(t) = 10+2t/√5 same apply to T(t) also

you see my point?

here are list of questions

1. Is pressure increasing or decreasing at the direction of ∇f(x)
2. How fast should i change my temperature and volume(suppose I am doing an experiment where i can change the rate of these 2 factors)
to give the maximum dP/dt at the given initial point T=T₀ and V=V₀
3. is there always a minimum dP/dt in the direction of -∇f(x)?

4.what does scaling the magnitude of ∇f(x) and U up and down mean(which of course, affect the value of dP/dt) to my experiment does it make any sense in physical point of view?