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Derivatives of multivariable functions

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Partial derivatives, introductionGraphical understanding of partial derivativesFormal definition of partial derivativesSymmetry of second partial derivatives
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GradientGradient and graphsGradient and contour mapsDirectional derivativeDirectional derivative, formal definitionDirectional derivatives and slopeWhy the gradient is the direction of steepest ascent
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Introduction to partial derivativesSecond partial derivativesThe gradientDirectional derivatives (introduction)Directional derivatives (going deeper)
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Vector-valued functions introVector-valued functions differentiationDifferential of a vector valued functionVector valued function derivative example
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Multivariable chain ruleMultivariable chain rule intuitionVector form of the multivariable chain ruleMultivariable chain rule and directional derivativesMore formal treatment of multivariable chain rule
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Curvature intuitionCurvature formula, part 1Curvature formula, part 2Curvature formula, part 3Curvature formula, part 4Curvature formula, part 5Curvature of a helix, part 1Curvature of a helix, part 2Curvature of a cycloid
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Computing the partial derivative of a vector-valued functionPartial derivative of a parametric surface, part 1Partial derivative of a parametric surface, part 2Partial derivatives of vector fieldsPartial derivatives of vector fields, component by component
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Derivatives of vector-valued functionsCurvatureMultivariable chain rule, simple versionPartial derivatives of parametric surfaces
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Divergence intuition, part 1Divergence intuition, part 2Divergence formula, part 1Divergence formula, part 2Divergence exampleDivergence notation
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2d curl intuition2d curl formula2d curl example2d curl nuanceDescribing rotation in 3d with a vector3d curl intuition, part 13d curl intuition, part 23d curl formula, part 13d curl formula, part 23d curl computation example
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DivergenceIntuition for divergence formulaCurl warmup, fluid rotation in two dimensionsCurl, fluid rotation in three dimensions
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Laplacian intuitionLaplacian computation exampleExplicit Laplacian formulaHarmonic Functions
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Jacobian prerequisite knowledgeLocal linearity for a multivariable functionThe Jacobian matrixComputing a Jacobian matrixThe Jacobian Determinant
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About this unit

What does it mean to take the derivative of a function whose input lives in multiple dimensions? What about when its output is a vector? Here we go over many different ways to extend the idea of a derivative to higher dimensions, including partial derivatives​, directional derivatives, the gradient, vector derivatives, divergence, curl, etc.