MAT1008 Fundamentals of Calculus
Directional Derivative, Gradient and
Lagrange Multiplier Method
Dr. Aswathy R K
Department of Mathematics
School of Advanced Sciences
Vellore Institute of Technology – Andhra Pradesh
Directional Derivatives
Rate of change in f (x, y) with respect to distance from a point (x0, y0) in some direction
Let u = u1i + u2 j be a unit vector in the desired direction.
(x, y)
s
(x0, y0)
x = x0 + su1
y = y0 +su2
s : Arc length parameter
s = 0 refers to (x0, y0)
s increases (x, y) moves away from (x0, y0) in the direction
of u
z = f (x, y) = f (x0 + su1, y0 + su2)
Directional Derivative, Gradient and Lagrange Multiplier Method
Directional Derivatives
Directional Derivative, Gradient and Lagrange Multiplier Method
Example
Directional Derivative, Gradient and Lagrange Multiplier Method
Directional Derivatives
Directional Derivative, Gradient and Lagrange Multiplier Method
Example
Directional Derivative, Gradient and Lagrange Multiplier Method
Gradient
Directional Derivative, Gradient and Lagrange Multiplier Method
Gradient
Directional Derivative, Gradient and Lagrange Multiplier Method
Example
Directional Derivative, Gradient and Lagrange Multiplier Method
Extremum Problems with Constrains / Constrained Extremum
Directional Derivative, Gradient and Lagrange Multiplier Method
Lagrange Multiplier
Directional Derivative, Gradient and Lagrange Multiplier Method
Lagrange Multiplier
Directional Derivative, Gradient and Lagrange Multiplier Method
Lagrange Multiplier Method
Directional Derivative, Gradient and Lagrange Multiplier Method
Example
Directional Derivative, Gradient and Lagrange Multiplier Method
Example
Directional Derivative, Gradient and Lagrange Multiplier Method
Example
Directional Derivative, Gradient and Lagrange Multiplier Method
Thank You…
Directional Derivative, Gradient and Lagrange Multiplier Method