Calculus/MVCalculus

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Expert: Alon Mandes - 8/18/2009


Question
Hi there if you could help me solve this that would be great.


The temperature of a star decays from the centre towards the surface as

T(x,y,z)= Toe^-(x^4+y^4+z^4)

Where To is the temperature in the centre and x,y,z are non dimensional Cartesian coordinates originating in the centre. The value of To is not specified. Find the rated of change of the temperature at an arbitrary point (xo,yo,,zo) in the x direction, y direction and z direction as expressed I terms of To, xo, yo and zo


thanks

Answer
Since T(x,y,z) is a 3D function , the derivative will be not only with respect to the variable
but also with respect to the direction specified as vector. It's called Directional derivative.
It's defined as :
∂T/∂v = ▼T•v . Where • is scalar product , & v is the direction vector .  ▼T is the Grad vector
operator [∂/∂x,∂/∂y,∂/∂z] . Our direction is along the x-axis, that means v=[x,0,0].
Therefore:
∂T/∂v=[∂/∂x,∂/∂y,∂/∂z]•[x,0,0]=∂T/∂x . Let's calculate this form :
∂T/∂x=To*4x³*e^-(x^4+y^4+z^4). Note that ∂T/∂x is itself a function of x,y&z. We are interested
in spicific location : point (xo,yo,zo) . Thus :
∂T/∂x                = To*4xo³*e^-(xo^4+yo^4+zo^4).
|at point xo,yo,zo)|

Alon.

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Alon Mandes

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Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

Experience

1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

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Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.

Education/Credentials
M.A in Mathematics & Bs.c in Electronics.

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