gradient of normal 1 Gradient of a function i e f x f x is not normal to the curve f x f x It is normal to the contour curve and tangent to the curve f x f x Consider a function z f x y z f x y then rewrite it as f x t y t c f x t y t c with some constant scalar c c for each level surface i e a surface with
I know what gradient vector or nabla F is and I know how to prove that it is orthogonal to the surface using calculation not intuitive In a particular case in which we have a three variable function I want to know why the What is the difference between the gradient of the tangent line and a normal vector of a curve I understand they mean different things but the equations are very similar For example I want to find the gradient and normal vector for the curve x 2 y 2 25 at the point 3 4 Won t I get the same vector for both
gradient of normal
gradient of normal
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3 The gradient is also supposed to be perpendicular to the tangent of a plane its normal vector This isn t true The gradient vector is perpendicular to the curve f x y 0 not perpendicular to the plane containing the curve Share Where v t r t v t r t What this shows is that any curve on the surface defined by g 0 g 0 has velocity perpendicular to the gradient of g g Being perpendicular to the velocity of any curve on the surface is exactly what we mean when we say that a vector is perpendicular to a surface Share
0 For your first question We say a vector v Rn is perpendicular to S Rn at p S if for every curve a a Rn s t Img S and 0 p it holds that 0 v 0 With this definition Suppose that f p 0 Let a a L f p with 0 p Notice that x L f p f x f p is constant First of all the concept of inward and outward normal makes sense only if the surface is closed But even for closed surfaces there is no relation The level curves f x y z C f x y z C and f x y z C f x y z C are the same but in one case the gradient is an outward normal and in the other an inward normal
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The gradient and the normal vector are closely related as they both represent the direction of steepest ascent or descent of a function The gradient is perpendicular to the level curves of a function while the normal vector is perpendicular to the surface of a function 3 R is a nxn matrix A is a nxm matrix b is a mx1 vector Are you saying it s not possible to find the gradient of this norm I know the least squares problem is supposed to correspond to normal equations and I was told that I could find the normal equations that the least square problem corresponded to by taking the gradient endgroup
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gradient of normal - 0 For your first question We say a vector v Rn is perpendicular to S Rn at p S if for every curve a a Rn s t Img S and 0 p it holds that 0 v 0 With this definition Suppose that f p 0 Let a a L f p with 0 p Notice that x L f p f x f p is constant