Partial Derivatives GATE 2017 :Calculus
Partial Derivatives GATE 2017 :Calculus: Engineering Mathematics
Full PDF: Partial Derivatives GATE 2017
Video Lecture Partial Derivatives GATE 2017 :Calculus: Engineering Mathematics
Partial derivatives are defined as derivatives of a function of multiple variables when all but the variable of interest are held fixed during the differentiation.
(1)
The above partial derivative is sometimes denoted 
for brevity.Partial derivatives can also be taken with respect to multiple variables, as denoted for examples
(2) 
(3) 
(4)
Such partial derivatives involving more than one variable are called mixed partial derivatives.For a "nice" two-dimensional function 
(i.e., one for which 
, 
, 
, 
, 
exist and are continuous in a neighborhood 
), then
(5)
More generally, for "nice" functions, mixed partial derivatives must be equal regardless of the order in which the differentiation is performed, so it also is true that
(6)
If the continuity requirement for mixed partials is dropped, it is possible to construct functions for which mixed partials are not equal. An example is the function
(7)
which has 
and 
(Wagon 1991). This function is depicted above and by Fischer (1986).Abramowitz and Stegun (1972) give finite difference versions for partial derivatives.A differential equation expressing one or more quantities in terms of partial derivatives is called a partial differential equation. Partial differential equations are extremely important in physics and engineering, and are in general difficult to solve.
Partial derivatives are defined as derivatives of a function of multiple variables when all but the variable of interest are held fixed during the differentiation.
| (1) |
The above partial derivative is sometimes denoted for brevity.
for brevity.Partial derivatives can also be taken with respect to multiple variables, as denoted for examples
| | | (2) |
| | | (3) |
| | | (4) |
Such partial derivatives involving more than one variable are called mixed partial derivatives.
For a "nice" two-dimensional function (i.e., one for which , , , , exist and are continuous in a neighborhood ), then
(i.e., one for which , , , , exist and are continuous in a neighborhood ), then | (5) |
More generally, for "nice" functions, mixed partial derivatives must be equal regardless of the order in which the differentiation is performed, so it also is true that
| (6) |
If the continuity requirement for mixed partials is dropped, it is possible to construct functions for which mixed partials are not equal. An example is the function
| (7) |
which has and (Wagon 1991). This function is depicted above and by Fischer (1986).
and (Wagon 1991). This function is depicted above and by Fischer (1986).Abramowitz and Stegun (1972) give finite difference versions for partial derivatives.
A differential equation expressing one or more quantities in terms of partial derivatives is called a partial differential equation. Partial differential equations are extremely important in physics and engineering, and are in general difficult to solve.
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