Gradient vectors

Gradient vectors

Lessons

Notes:

Gradient Vector

The gradient vector (denoted as f\nabla f) is a vector where all the components are partial derivatives of the function in respect to each variable. Also known as the direction with the greatest increase of ff

For example, consider the function f(x,y,z)f(x,y,z). Then,

f=<fx,fy,fz>\nabla f = \lt f_x, f_y, f_z\gt

If you want to find the gradient of a specific point (x0,y0,z0)(x_0, y_0, z_0), then

f(x0,y0,z0)=<fx(x0,y0,z0),fy(x0,y0,z0),fz(x0,y0,z0)> \nabla f(x_0, y_0, z_0)= \lt f_x(x_0, y_0, z_0),f_y(x_0, y_0, z_0), f_z(x_0, y_0, z_0)\gt


Finding the Tangent Plane with Gradient

Gradients are useful for finding the tangent plane.

Recall that the equation of a plane is:

a(xx0)+b(yy0)+c(zz0)=0 a(x-x_0)+b(y-y_0)+c(z-z_0)=0

The gradient vector is actually the normal vector that is orthogonal to the tangent plane at (x0,y0,z0)(x_0, y_0, z_0). So that means:

a=fx(x0,y0,z0)a=f_x(x_0, y_0, z_0)
b=fy(x0,y0,z0)b=f_y(x_0, y_0, z_0)
c=fz(x0,y0,z0)c=f_z(x_0, y_0, z_0)


Finding the Normal Line with Gradient

There are times in which instead of finding the normal vector, we want the normal line. Recall that the formula for a vector equation is:

r(t)=<x0,y0,z0>+t<a,b,c>r(t)= \lt x_0, y_0, z_0\gt+ t\lt a,b,c\gt

Since the gradient is the direction of the vector, and we already have an initial point (x0,y0,z0)(x_0, y_0, z_0), then the normal line is:

r(t)=<x0,y0,z0>+tf(x0,y0,z0)r(t)= \lt x_0, y_0, z_0\gt + t \nabla f(x_0, y_0, z_0)

  • Introduction
    Gradient Vectors Overview:
    a)
    • Gradient vector = f\nabla f
    • Direction with the greatest increase of ff
    • Components are partial derivatives <fx,fy,fz>\to \lt f_x,f_y, f_z\gt
    • Gradient vector at a point =f(x0,y0,z0)=\nabla f(x_0, y_0, z_0)
    • An Example

    b)
    Finding the Tangent Plane with Gradient
    • Can use Gradient to find tangent planes
    • Recall equation of a plane
    • Gradient = normal vector orthogonal to tangent plane
    • An Example

    c)
    Finding the Normal Line with Gradient
    • Recall vector equations
    • Gradient = direction of vector
    • r(t)=<x0,y0,z0>+tf(x0,y0,z0)r(t)= \lt x_0, y_0, z_0\gt+ t \nabla f(x_0, y_0, z_0)
    • An example