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Lectures 9 and 10 Directional Derivatives and Gradient. Taylor Expansions Calculus II Topic 1: Differential Calculus in Several Variables

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Paths of Steepest Descent

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Directional Derivatives (I)

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Directional Derivatives (II)

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Directional Derivatives (III)

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Finding the Directional Derivative using the Definition

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Directional Derivative: General Notation

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Gradient Vector and Directional Derivative

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Finding Directional Derivative using Gradient

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Finding Directional Derivative using Gradient

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Directional Derivative as the Slope of the Tangent Line in direction of u

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Directional Derivative as the Slope of the Tangent Line in direction of u

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Properties of Directional Derivatives and Gradients

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Directions of Steepest Ascent and Descent: General Case

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Directions of Steepest Ascent and Descent: Example

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Gradient is normal to Level Curves: Example

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Gradient is normal to Level Curves

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Why the water flow paths are perpendicular to contour lines ?

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Path of Steepest Descent: Example

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Application: Tangent to a Plane Curve

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Tangent to a Plane Curve

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Gradient in Physics: Conservative Fields. The Gravitational Force is a Gradient or Conservative Field

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Gradient in Physics: Conservative Fields. The Electrical Field is a Gradient or Conservative Field

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Gradient and Directional Derivative in 3D

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Gradient is normal to Level Surfaces in 3D

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Example in 3D

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Calculus with Gradients

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Tangent Plane revisited

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Tangent Plane revisited: Example

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Building Tangent Plane from two Curves on the Surface ( without knowing the Surface !!!! )

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Tangent to a 3D Curve

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Directional Differentials

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Directional Differential: Example

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Jacobian Matrix

=

=

=

𝐼𝑓 𝑚 = 1 𝑠𝑐𝑎𝑙𝑎𝑟 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 , 𝑡ℎ𝑒𝑛 𝑫𝒇 = (𝒈𝒓𝒂𝒅 𝑓)𝑇 = Calculus II

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𝜕𝑓 𝜕𝑥1

𝜕𝑓

… 𝜕𝑥

𝑛

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Hessian Matrix

If đ?’‡ has continuous second partial derivatives, then Schwarz (Claireaut) Theorem holds and Hessian Matrix đ?‘Ż đ?’‡ is symmetric.

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Taylor Expansions: The 1D Case

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Taylor Expansions of sin(x)

Degree 1, 3, 5, 7, 9, 11, 13 Calculus II

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Taylor Expansions of exp(x)

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Taylor Expansion: Multidimensional Case

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Taylor Theorem

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Error in Taylor Expansions

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High Order Taylor Expansions

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Local Approximation of Function by a Plane (first order) or a Paraboloid (second order)

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2D Example

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1.9_1.10_Derivadas_Direccionales_Gradiente_Taylor