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The heat diffusion equation Let $T(t, x, y, z)$ be a function with continuous second derivatives giving the temperature at time $t$ at the point $(x, y, z)$ of a solid occupying a region $D$ in space. If the solid's heat capacity and mass density are denoted by the constants $c$ and $\rho,$ respectively, the quantity $c \rho T$ is called the solid's heat energy per unit volume.a. Explain why $-\nabla T$ points in the direction of heat flow.b. Let $-k \nabla T$ denote the energy flux vector. (Here the constant$k$ is called the conductivity.) Assuming the Law of Conservation of Mass with $-k \nabla T=\mathbf{v}$ and $c \rho T=p$ in Exercise 31 derive the diffusion (heat) equation$$\frac{\partial T}{\partial t}=K \nabla^{2} T$$where $K=k /(c \rho)>0$ is the diffusivity constant. (Notice that if $T(t, x)$ represents the temperature at time $t$ at position $x$ in a uniform conducting rod with perfectly insulated sides, then $\nabla^{2} T=\partial^{2} T / \partial x^{2}$ and the diffusion equation reduces to the one-dimensional heat equation in Chapter 14 's Additional Exercises.)

Calculus 3

Chapter 16

Integrals and Vector Fields

Section 8

The Divergence Theorem and a Unified Theory

Vector Functions

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Boston College

Lectures

03:04

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x. The input of a function is called the argument and the output is called the value. The set of all permitted inputs is called the domain of the function. Similarly, the set of all permissible outputs is called the codomain. The most common symbols used to represent functions in mathematics are f and g. The set of all possible values of a function is called the image of the function, while the set of all functions from a set "A" to a set "B" is called the set of "B"-valued functions or the function space "B"["A"].

08:32

In mathematics, vector calculus is an important part of differential geometry, together with differential topology and differential geometry. It is also a tool used in many parts of physics. It is a collection of techniques to describe and study the properties of vector fields. It is a broad and deep subject that involves many different mathematical techniques.

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