How Does Temperature Change Along a Helical Path?

[jonroberts74]

the temperature at a point in space is $$T(x,y,z) = x^2+y^2+z^2$$

and there is a particle traveling along the helix given by

$$\sigma (t) =(cos(t),sin(t),t)$$

a) find $$T'(t)$$

$$T'(t) = \frac{\partial T}{\partial x} \frac{dx}{dt} + \frac{\partial T}{\partial y}\frac{dy}{dt} + \frac{\partial T}{\partial z} \frac{dz}{dt}$$

$$= -2cos(t)sin(t) + 2sin(t)cos(t) +2t = 2t$$

b) find the temperature at time $$t = \frac{\pi}{2} + 0.01$$

$$= cos^2 (t) + sin^2 (t) + t^2$$

evaluated at the given t

$$\approx 3.50$$how does this look?

thanks!

 

[verty]

The last answer doesn't look right, I mean 1 + t^2, $\pi^2 \over 4$ should be about 2.5, not 3.5.

 

[jonroberts74]

The last answer doesn't look right, I mean 1 + t^2, $\pi^2 \over 4$ should be about 2.5, not 3.5.

$$1+\left( \frac{\pi}{2} + 0.01\right)^2 = 3.49891702681$$

 

[SammyS]

the temperature at a point in space is $$T(x,y,z) = x^2+y^2+z^2$$

and there is a particle traveling along the helix given by

$$\sigma (t) =(cos(t),sin(t),t)$$

a) find $$T'(t)$$

$$T'(t) = \frac{\partial T}{\partial x} \frac{dx}{dt} + \frac{\partial T}{\partial y}\frac{dy}{dt} + \frac{\partial T}{\partial z} \frac{dz}{dt}$$

$$= -2cos(t)sin(t) + 2sin(t)cos(t) +2t = 2t$$

b) find the temperature at time $$t = \frac{\pi}{2} + 0.01$$

$$= cos^2 (t) + sin^2 (t) + t^2$$

evaluated at the given t

$$\approx 3.50$$


how does this look?

thanks!

It looks good !