Math 210 Lecture 7: Directional derivative,
gradient and linear approximation
Daniel Ingebretson
University of Illinois at Chicago
June 26, 2024
,The gradient
Let’s rewrite the formula for the directional derivative:
Du f (a, b) = fx (a, b) u1 + fy (a, b) u2
.
,The gradient
Let’s rewrite the formula for the directional derivative:
Du f (a, b) = fx (a, b) u1 + fy (a, b) u2
= fx (a, b), fy (a, b) · ⟨u1 , u2 ⟩.
, The gradient
Let’s rewrite the formula for the directional derivative:
Du f (a, b) = fx (a, b) u1 + fy (a, b) u2
= fx (a, b), fy (a, b) · ⟨u1 , u2 ⟩.
Definition
The vector
∇f (a, b) = fx (a, b), fy (a, b)
is called the gradient of f at (a, b).
gradient and linear approximation
Daniel Ingebretson
University of Illinois at Chicago
June 26, 2024
,The gradient
Let’s rewrite the formula for the directional derivative:
Du f (a, b) = fx (a, b) u1 + fy (a, b) u2
.
,The gradient
Let’s rewrite the formula for the directional derivative:
Du f (a, b) = fx (a, b) u1 + fy (a, b) u2
= fx (a, b), fy (a, b) · ⟨u1 , u2 ⟩.
, The gradient
Let’s rewrite the formula for the directional derivative:
Du f (a, b) = fx (a, b) u1 + fy (a, b) u2
= fx (a, b), fy (a, b) · ⟨u1 , u2 ⟩.
Definition
The vector
∇f (a, b) = fx (a, b), fy (a, b)
is called the gradient of f at (a, b).
