Book contents
- Frontmatter
- Contents
- Preface to the second edition
- Preface to the first edition
- 1 Preliminary algebra
- 2 Preliminary calculus
- 3 Complex numbers and hyperbolic functions
- 4 Series and limits
- 5 Partial differentiation
- 6 Multiple integrals
- 7 Vector algebra
- 8 Matrices and vector spaces
- 9 Normal modes
- 10 Vector calculus
- 11 Line, surface and volume integrals
- 12 Fourier series
- 13 Integral transforms
- 14 First-order ordinary differential equations
- 15 Higher-order ordinary differential equations
- 16 Series solutions of ordinary differential equations
- 17 Eigenfunction methods for differential equations
- 18 Partial differential equations: general and particular solutions
- 19 Partial differential equations: separation of variables and other methods
- 20 Complex variables
- 21 Tensors
- 22 Calculus of variations
- 23 Integral equations
- 24 Group theory
- 25 Representation theory
- 26 Probability
- 27 Statistics
- 28 Numerical methods
- Appendix Gamma, beta and error functions
- Index
5 - Partial differentiation
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface to the second edition
- Preface to the first edition
- 1 Preliminary algebra
- 2 Preliminary calculus
- 3 Complex numbers and hyperbolic functions
- 4 Series and limits
- 5 Partial differentiation
- 6 Multiple integrals
- 7 Vector algebra
- 8 Matrices and vector spaces
- 9 Normal modes
- 10 Vector calculus
- 11 Line, surface and volume integrals
- 12 Fourier series
- 13 Integral transforms
- 14 First-order ordinary differential equations
- 15 Higher-order ordinary differential equations
- 16 Series solutions of ordinary differential equations
- 17 Eigenfunction methods for differential equations
- 18 Partial differential equations: general and particular solutions
- 19 Partial differential equations: separation of variables and other methods
- 20 Complex variables
- 21 Tensors
- 22 Calculus of variations
- 23 Integral equations
- 24 Group theory
- 25 Representation theory
- 26 Probability
- 27 Statistics
- 28 Numerical methods
- Appendix Gamma, beta and error functions
- Index
Summary
In chapter 2, we discussed functions f of only one variable x, which were usually written f(x). Certain constants and parameters may also have appeared in the definition of f, e.g. f(x) = ax + 2 contains the constant 2 and the parameter a, but only x was considered as a variable and only the derivatives f(n)(x) = dnf/dxn were defined.
However, we may equally well consider functions that depend on more than one variable, e.g. the function f(x, y) = x2 + 3xy, which depends on the two variables x and y. For any pair of values x, y, the function f(x, y) has a well-defined value, e.g. f(2, 3) = 22. This notion can clearly be extended to functions dependent on more than two variables. For the n-variable case, we write f(x1, x2, …, xn) for a function that depends on the variables x1, x2, …, xn. When n = 2, x1 and x2 correspond to the variables x and y used above.
Functions of one variable, like f(x), can be represented by a graph on a plane sheet of paper, and it is apparent that functions of two variables can, with little effort, be represented by a surface in three-dimensional space. Thus, we may also picture f(x, y) as describing the variation of height with position in a mountainous landscape. Functions of many variables, however, are usually very difficult to visualise and so the preliminary discussion in this chapter will concentrate on functions of just two variables.
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- Chapter
- Information
- Mathematical Methods for Physics and EngineeringA Comprehensive Guide, pp. 154 - 189Publisher: Cambridge University PressPrint publication year: 2002