Lecture 12 Chain rule f x y z and r t x t y t z t Gradient of f Example f x y z ax by cz Example g x y x y Why does the gradient give a vector perpendicular to the level set The gradient must be perpendicular to any curve in the level set Suppose r is inside f d We know that Application Tangent planes This means that the gradient is perpendicular to the level set QED F x y z x y z Find the tangent plane to the level set f 169 at 3 4 12 Find the normal direct to the tangent plane using the gradient Compute the gradient of f Any scalar multiple is also normal to the plane i e 3 4 12 Find the tangent plane Finding the normal to a graph z g x y De ne a function F x y z z g x y F g g 1 Find the tangent plane to xyz 1 at 1 2 1 2 f xyz level set f 1 Differential Derivative z f x y Pick direction How does f change in the direction u
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