Directional derivatives

Directional derivatives

Lessons

Notes:


Suppose we have a vector v=<v1,v2>\vec{v} =\lt v_1, v_2\gt. The unit vector will be:

v=1v12+v22<v1,v2>\vec{v} = \frac{1}{\sqrt{v^2_1 + v^2_2}} \lt v_1,v_2\gt

Suppose we have a vector v=<v1,v2,v3>\vec{v} =\lt v_1, v_2,v_3\gt. The unit vector will be:

v=1v12+v22+v32<v1,v2,v3> \vec{v} = \frac{1}{ \sqrt{v_1^2 + v_2^2 + v_3^2} } <v_1,v_2,v_3>

When given an angle of a direction (θ\theta ), we say that the unit vector (that points to the direction) is:

u=<cosθ,sinθ>\vec{u} = \lt \cos \theta, \sin \theta \gt


Directional Derivatives of 2 Variable Functions
A Directional Derivative is the rate of change (of xx and yy) of a function at a point P=(x0,y0,z0)P=(x_0,y_0,z_0), at the direction of the unit vector

Suppose there is a 2-variable function z=f(x,y)z=f(x,y). Then the directional derivative is:

Duf(x,y)=fx(x,y)a+fy(x,y)bD_{\vec{u}}f(x,y) = f_x(x,y)a + f_y(x,y)b

where the u=<a,b>\vec{u} = <a,b> is the unit vector that points in the direction of change. Directional Derivatives of 3 Variable Functions
Suppose there is a 3-variable function w=f(x,y,z)w=f(x,y,z). Then the directional derivative is:

Duf(x,y,z)=fx(x,y,z)a+fy(x,y,z)b+fz(x,y,z)cD_{\vec{u}}f(x,y,z) = f_x(x,y,z)a + f_y(x,y,z)b + f_z(x,y,z)c

where the u=<a,b,c>\vec{u} = <a,b,c> is the unit vector that points in the direction of change.
  • Introduction
    Directional Derivatives Overview:
    a)
    Things to Know Before Knowing Directional Derivatives
    • Calculating unit vectors
    • An example
    • Angle to a unit vector
    • An example

    b)
    Directional Derivatives of 2 Variable Functions
    • The rate of change of xx and yy
    • Duf(x,y)=fx(x,y)a+fy(x,y)bD_{\vec{u}}f(x,y) = f_x(x,y)a + f_y(x,y)b
    • An example

    c)
    Directional Derivatives of 3 Variable Functions
    • The rate of change of xx and yy
    • Duf(x,y,z)=fx(x,y,z)a+fy(x,y,z)b+fz(x,y,z)cD_{\vec{u}}f(x,y,z) = f_x(x,y,z)a + f_y(x,y,z)b + f_z(x,y,z)c
    • An example


  • 1.
    Finding the Unit Vector & Angle of Direction
    Find the unit vector of v=<5,2>.\vec{v} =\lt 5, -2\gt.

  • 2.
    Find the unit vector of v=<1,3,5>.\vec{v} =\lt -1, 3, 5\gt.

  • 3.
    Find the unit vector, given that the unit vector is in the direction of θ=π3\theta=\frac{\pi}{3}.

  • 4.
    Finding the Directional Derivative of 2 Variable Functions
    Find the direction derivative of z=x2+2yz=\sqrt{x^2+2y} at any given point, in the direction of v=<1,3> \vec{v} = \lt 1, 3\gt

  • 5.
    Find the direction derivative of z=xyln(xy)z=xy \ln (\frac{x}{y}) at any given point, where the direction of the unit vector is at θ=π6\theta=\frac{\pi}{6} .

  • 6.
    Finding the Directional Derivative of 3 Variable Functions
    Find the direction derivative of f(x,y,z)=xy3+yz2f(x,y,z)=xy^3+yz^2 in the direction of v=<1,2,4>\vec{v} =\lt 1, 2, 4\gt.

  • 7.
    Find the direction derivative of f(x,y,z)=ln(x)eyzf(x,y,z)=\ln (x)e^{yz} in the direction of v=<3,1,2>\vec{v} =\lt -3, 1, 2\gt .