Gradient

Theorem:
Gradient and Level Curves
If $f(x,y)$ is differentiable at the point $(x_0,y_0)$, then the gradient of $f$ at $(x_0,y_0)$ is perpendicular to the level curve of $f$ at $(x_0,y_0)$.
Loading Plotly...

Example

Consider the function $w = f(x,y,z) = x^2+y^2+z^2$. The level sets of this function are spheres centered at the origin.

Loading Plotly...
Directional Derivative

Example

Determine the rate of change of the function $f(x,y) = x^2 y^2 + 2xy^3 +y^4$ at the point $(1,2)$ in the direction of the vector $\mathbf{v}$.

Theorem:
Geometric Properties of the Gradient Vector
  1. At a point $(x_1, \ldots, x_n)$, the gradient vector $\nabla f(x_1, \ldots, x_n)$ points in the direction of the greatest rate of increase of $f$. The maximum rate of increase is $\Vert \nabla f(x_1, \ldots, x_n) \Vert$.
  2. At a point $(x_1, \ldots, x_n)$, the negative of the gradient vector $-\nabla f(x_1, \ldots, x_n)$ points in the direction of the greatest rate of decrease of $f$. The maximum rate of decrease is $-\Vert \nabla f(x_1, \ldots, x_n) \Vert$.
  3. The rate of change of $f(x_1, \ldots, x_n)$ is zero in the directions tangent to the level set of $f$ at $(x_1, \ldots, x_n)$.

Determine the direction of steepest ascent for the function $f(x,y) = x^2 y^2 + 2xy^3 +y^4$ at the point $(1,2)$ in the direction of the vector $\mathbf{v}$.

Mountain stream

You are standing besides a stream in the Laurentian mointains. The height of the terrain (in meters) is given by $$h(x,y) = \frac{1000}{2x^2+y^2+3}$$ Here $x$ and $y$ (in kilometers) are the coorinates of the point on your map. You are at the point $(2,3)$.

Loading Plotly...