Mathematical Analysis 2. Lecture 1. Functions of two or more independent variables. Differential

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Mathematical Analysis 2
Lecture 1 - Functions of two or more independent
variables. Differential. Partial derivatives with constrained
variables
Aigul Myrzagaliyeva
[email protected]
Astana IT University

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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

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Multivariable functions

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Multivariable functions

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Interior and boundary points, open and closed region

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Interior and boundary points, open and closed region

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Interior and boundary points, open and closed region

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Interior and boundary points, open and closed region

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Interior and boundary points,open and closed region

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Level Curve, Graph of function

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Level Curve, Graph of function

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Level surface,

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Limits of multivariable functions

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Limits along smooth curves

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Limits of multivariable functions

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Limits of multivariable functions

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Limits along smooth curves

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Continuity of multivariable functions
Theorem:

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Continuity of multivariable functions

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Continuity of multivariable functions

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Partial differentiation

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Partial differentiation

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Partial differentiation

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Partial differentiation

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Partial differentiation

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Partial differentiation

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Partial differentiation

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Partial differentiation

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Partial differentiation
f 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

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Partial differentiation

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Partial differentiation

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Chain rules for partial differentiation
Definition (chain rules):

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Chain rules for partial differentiation

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Chain rules for partial differentiation
Definition (chain rules) for partial derivatives:

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Chain rules for partial differentiation

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Chain rules for partial differentiation

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Chain rules for partial differentiation

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Chain rules for partial differentiation

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Lecture Summary
• Multivariable functions
• Limits of multivariable functions
• Limits along smooth curves
• Continuity of multivariable functions
• Partial differentiation
• Chain rules for partial differentiation