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
=
=
=
πΌπ π = 1 π πππππ ππ’πππ‘πππ , π‘βππ π«π = (ππππ π)π = Calculus II
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ππ ππ₯1
ππ
β¦ ππ₯
π
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Hessian Matrix
If Δ?’‡ has continuous second partial derivatives, then Schwarz (Claireaut) Theorem holds and Hessian Matrix Δ?‘Ε» Δ?’‡ 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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