heat equation

[shoogar]

why does the one-dimensional heat equation for temperature distribution contain a second derivative of the spatial variable?

[Integral]

Because it wouldn't be the heat equation if it didn't?

Perhaps you need to refer to a text on PDEs for a derivation.

[Edgardo]

Have a look at these derviations (1 (banach.millersville.edu/~bob/math467/HeatEquation3D.pdf) and 2 (http://planetmath.org/encyclopedia/DerivationOfHeatEquation.html)). You consider the flux which is given by a surface integral over grad(u). This surface integral is turned into volume integral by using the divergence theorem (http://en.wikipedia.org/wiki/Divergence_theorem). The divergence of grad(u) is laplace(u), or in one dimension u_xx.
To understand the article it is helpful to know about the following topics:

1) Divergence theorem:
- Examples for the divergence theorem (also known as Gauss theorem) can be found here:
Gauss Theorem Examples (http://math.bard.edu/~mbelk/math601/GaussExamples.pdf)
Paul's Online Math Notes (http://tutorial.math.lamar.edu/Classes/CalcIII/DivergenceTheorem.aspx)

2) Specific heat capacity:
- Specific heat capacity (http://www.taftan.com/thermodynamics/CP.HTM)

3) Gradient:
- Lecture by Edward Frenkel (Math Berkeley)
At 3:56 he gives an intuitive explanation of the gradient.
Mathematics - Multivariable Calculus - Lecture 12 (http://www.youtube.com/watch?v=7cPcutRLLXE)
- Videos by Salman Khan:
Gradient 1 (http://www.youtube.com/watch?v=U7HQ_G_N6vo)
Gradient of a scalar field (http://www.youtube.com/watch?v=OB8b8aDGLgE)
In the second video he shows the gradient of a scalar field T(x,y,z) defined in 3 dimensional space.

[chwala]

kindly refer to the pde notes online .............its like asking why does the wave equation have u{tt} - c^2 u{xx}=2t=f(x,t) have this form that leads to d alemberts equation