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18.02 Multivariable Calculus
Fall
200
7
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P. Partial Differential Equations
An important application of the higher partial derivatives is that they are used in partial
differential equations to express some laws of physics which are basic to most science and
engineering subjects. In this section, we will give examples of a few such equations. The
reason is partly cultural, so you meet these equations early and learn to recognize them,
and partly technical: to give you a little more practice with the chain rule and computing
higher derivatives.
A partial differential equation, PDE for short, is an equation involving some unknown
function of several variables and one or more of its partial derivatives. For example,
is such an equation. Evidently here the unknown function is a function of two variables
we infer this from the equation, since only x and y occur in it
as
independent variables. In
general a solution of a partial differential equation is a differentiable function that satisfies
it. In the above example, the functions
w
=
xnyn
any
n
all are solutions to the equation. In general, PDE's have many solutions, far too many
to find all of them. The problem is always to find the one solution satisfying some extra
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 Spring '08
 Auroux
 Differential Equations, Equations, Derivative, Multivariable Calculus, Partial Differential Equations, Laplace, Partial differential equation, temperature distribution

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