Lectures 5 and 6 Partial Derivatives Tangent Planes and Differentials Calculus II Topic 1: Differential Calculus in Several Variables
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Functions of Several Variables: Different slopes depending on the direction
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Watching a two variables Function as a function of «x» only
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Watching a two variables Function as a function of «y» only
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Partial Derivatives: Definition
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Geometrical Interpretation (1 of 2)
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Geometrical Interpretation (2 of 2)
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Partial Derivatives in General
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Partial Derivatives: Usual Notations
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Example: Finding Partial Derivatives at a Point
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Example: Finding a Partial Derivative as a Function
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Geometrical Application
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Functions with more than two independent variables
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Physical Application
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Physical Application
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Existence of Partial Derivatives ≠> Continuity
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Continuous Partial Derivatives ≠> Continuity
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Second-Order Partial Derivatives
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The Schwarz (Claireaut) Theorem
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Finite Increments Theorem
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Differentiability
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Tangent Plane to a Surface
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Tangent Plane to a Surface
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Exercises
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Exercise
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Differentiability and Tangent Plane
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Singularities
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Tangent Plane as Linear Approximation
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Tangent Plane as Linear Approximation
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Linearization
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Differentiability
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Approximation by Linearization
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The Error in the Linear Approximation
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Example of Linearization
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The Differential
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Relationship between the Differential and the Increment
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Applications of the Differential
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Exercise
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Differentiability of Vectorial Functions of several variables
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Homework
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