Temperature at (x,y,z)

  • #1
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the temperature at a point in space is [tex]T(x,y,z) = x^2+y^2+z^2[/tex]

and there is a particle traveling along the helix given by

[tex]\sigma (t) =(cos(t),sin(t),t)[/tex]

a) find [tex]T'(t)[/tex]

[tex]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}[/tex]

[tex] = -2cos(t)sin(t) + 2sin(t)cos(t) +2t = 2t [/tex]

b) find the temperature at time [tex] t = \frac{\pi}{2} + 0.01[/tex]

[tex] = cos^2 (t) + sin^2 (t) + t^2[/tex]

evaluated at the given t

[tex]\approx 3.50 [/tex]


how does this look?

thanks!
 

Answers and Replies

  • #2
verty
Homework Helper
2,182
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The last answer doesn't look right, I mean 1 + t^2, ##\pi^2 \over 4## should be about 2.5, not 3.5.
 
  • #3
189
0
The last answer doesn't look right, I mean 1 + t^2, ##\pi^2 \over 4## should be about 2.5, not 3.5.


[tex]1+\left( \frac{\pi}{2} + 0.01\right)^2 = 3.49891702681[/tex]
 
  • #4
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,379
1,039
the temperature at a point in space is [tex]T(x,y,z) = x^2+y^2+z^2[/tex]

and there is a particle traveling along the helix given by

[tex]\sigma (t) =(cos(t),sin(t),t)[/tex]

a) find [tex]T'(t)[/tex]

[tex]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}[/tex]

[tex] = -2cos(t)sin(t) + 2sin(t)cos(t) +2t = 2t [/tex]

b) find the temperature at time [tex] t = \frac{\pi}{2} + 0.01[/tex]

[tex] = cos^2 (t) + sin^2 (t) + t^2[/tex]

evaluated at the given t

[tex]\approx 3.50 [/tex]


how does this look?

thanks!
It looks good !
 

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