5.1 Definition of the partial derivative

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Chapter 5 Partial differentiation
5.1 Definition of the partial derivative

• the partial derivative of f(x,y) with respect to
  x and y are

• for general n-variable

• second partial derivatives of two-variable
  function f(x,y)

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Chapter 5 Partial differentiation
5.2 The total differential and total derivative
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Chapter 5 Partial differentiation
Ex Find the total derivative of
with respect to , given
that
5.3 Exact and inexact differentials

• If a function can be obtained by directly
  integrating its total differential, the
  differential of function f is called exact
  differential, whereas those that do not are
  inexact differential.

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Chapter 5 Partial differentiation
Ex Show that the differential xdy3ydx is
inexact.

• Inexact differential can be made exact by
  multiplying a suitable function called an
  integrating factor

Properties of exact differentials
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Chapter 5 Partial differentiation

• for n variables

Ex Show that (yz)dxxdyxdz is an exact
differential
5.4 Useful theorems of partial differentiation
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Chapter 5 Partial differentiation
5.5 The chain rule
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Chapter 5 Partial differentiation
5.6 Change of variables
Ex Polar coordinates ? and ?, Cartesian
coordinates x and y, x?cosf, y?sinf,
transform
into one in ? and f
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Chapter 5 Partial differentiation
5.7 Taylor