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Mathematical Analysis 2. Lecture 1. Functions of two or more independent variables. Differential
1.
Mathematical Analysis 2Lecture 1 - Functions of two or more independent
variables. Differential. Partial derivatives with constrained
variables
Aigul Myrzagaliyeva
[email protected]
Astana IT University
2.
Lecture overview• Multivariable functions
• Interior and boundary points, open and closed region
• Level Curve, Graph of function
• Limits of multivariable functions
• Continuity of multivariable functions
• Partial differentiation
• Chain rules for partial differentiation
3.
Multivariable functions4.
Multivariable functions5.
Interior and boundary points, open and closed region6.
Interior and boundary points, open and closed region7.
Interior and boundary points, open and closed region8.
Interior and boundary points, open and closed region9.
Interior and boundary points,open and closed region10.
Level Curve, Graph of function11.
Level Curve, Graph of function12.
Level surface,13.
Limits of multivariable functions14.
Limits along smooth curves15.
Limits of multivariable functions16.
17.
Limits of multivariable functions18.
19.
Limits along smooth curves20.
Continuity of multivariable functionsTheorem:
21.
Continuity of multivariable functions22.
Continuity of multivariable functions23.
Partial differentiation24.
Partial differentiation25.
Partial differentiation26.
Partial differentiation27.
Partial differentiation28.
Partial differentiation29.
Partial differentiation30.
Partial differentiation31.
Partial differentiationf x, y x cos y ye x
df
cos y ye x
dx
f x, y x cos y ye x
d df
d
cos y ye x
dy dx dy
df
x sin y e x
dy
d df
d
x sin y e x
dx dy dx
d2 f
sin y e x
dy dx
d2 f
sin y e x
dx dy
f y x sin y e x
f xy sin y e x
32.
Partial differentiation33.
Partial differentiation34.
Chain rules for partial differentiationDefinition (chain rules):
35.
Chain rules for partial differentiation36.
Chain rules for partial differentiationDefinition (chain rules) for partial derivatives:
37.
Chain rules for partial differentiation38.
Chain rules for partial differentiation39.
Chain rules for partial differentiation40.
Chain rules for partial differentiation41.
Lecture Summary• Multivariable functions
• Limits of multivariable functions
• Limits along smooth curves
• Continuity of multivariable functions
• Partial differentiation
• Chain rules for partial differentiation