# Temperature at (x,y,z)

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

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
Staff Emeritus
Homework Helper
Gold Member
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 !

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