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MATH
[1251]

Theorems from Derivatives and level-set geometry

All theorems from Derivatives and level-set geometry.

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Gradient normal to a level set

Let F:D→RF:D\to\mathbb{R} be C1C^1 on an open neighbourhood of x0\mathbf{x}_0 and let

S={x∈D:F(x)=c}.S=\{\mathbf{x}\in D:F(\mathbf{x})=c\}.

If x0∈S\mathbf{x}_0\in S and ∇F(x0)≠0\nabla F(\mathbf{x}_0)\ne\mathbf{0}, then ∇F(x0)\nabla F(\mathbf{x}_0) is normal to SS at x0\mathbf{x}_0.

The non-zero-gradient condition makes the level set regular at the point. When the gradient vanishes, the level set may have a corner, crossing or other singular behaviour for which no single gradient direction gives a normal.

  1. [1]:Gradient
  2. [2]:Level set
  3. [3]:Lagrange multipliers
  4. [4]:The closest point on a plane
  5. [5]:Constrained extrema
  6. [6]:Proof
  7. 5 more

Tangent plane to a graph

If ff is differentiable at a=(x0,y0)\mathbf{a}=(x_0,y_0) and z0=f(x0,y0)z_0=f(x_0,y_0), then the tangent plane to z=f(x,y)z=f(x,y) at (x0,y0,z0)(x_0,y_0,z_0) is

z=f(x0,y0)+fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0).z =f(x_0,y_0) +f_x(x_0,y_0)(x-x_0) +f_y(x_0,y_0)(y-y_0).

The point-normal form uses the normal (−fx(a),−fy(a),1)(-f_x(\mathbf{a}),-f_y(\mathbf{a}),1). The formula is also the first-order Taylor approximation, so the geometric tangent plane and the analytic linear model are the same object.

  1. [1]:Gradient normal to a level set
  2. [2]:Proof
  3. [3]:Approximate a function linearly

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Gradient normal to a level setTangent plane to a graph