Lecture Notes of

Thomas’ Calculus

Section 11.5

Directional Derivatives, Gradient Vectors and Tangent Planes

Instructor: Dr. ZHANG Zhengru Lecture Notes of Section 11.5 Page 1

OUTLINE

• Directional Derivatives in the Plane

• Interpretation of the Directional Derivative

• Calculation

• Properties of Directional Derivatives

• Gradients and Tangents to Level Curves

• Algebra Rules for Gradients

• Increments and Distances

• Functions of Three Variables

• Tangent Panes and Normal Lines

• Planes Tangent to a Surface z = f(x, y)

• Other Applications

Why introduce the directional derivatives?

• Let’s start from the derivative of single variable functions. Consider y = f(x), its ′ ′ derivative f (x0) implies the rate of change of f. f (x0) > 0 means f increasing as x ′ becomes large. f (x0) < 0 means f decreasing as x becomes large.

• Consider two-variable function f(x, y). The partial derivative fx(x0,y0) implies the

rate of change in the direction of~i while keep y as a constant y0. fx(x0,y0) > 0 implies increasing in x direction. Another interpretation, the vertical plane passes through

(x0,y0) and parallel to x-axis intersects the surface f(x, y) in the curve C. fx(x0,y0)

means the slope of the curve C. Similarly, The partial derivative fy(x0,y0) implies the

rate of change in the direction of ~j while keep x as a constant x0.

• It is natural to consider the rate of change of f(x, y) in any direction ~u instead of ~i and ~j. Therefore, directional derivatives should be introduced. ... Lecture Notes of Section 11.5 Page 2

Definition 1 The derivative of f at P0(x0,y0) in the direction of the unit vector

~u = u1~i + u2~j is the number

df f(x0 + su1,y0 + su2) − f(x0,y0) = lim , ds s→0 s  ~u,P0 provided the limit exists.

Calculation of Directional Derivatives

Consider f(x, y), P0(x0,y0), ~u = u1~i + u2~j

x = x0 + su1 y = y0 + su2 then df ∂f dx ∂f dy ∂f ∂f = + = · u1 + · u2 ds ∂x dt ∂y dt ∂x ∂y  ~u,P0  P0  P0  P0  P0 ∂f ∂f = ~i + ~j · (u1~i + u2~j)=(∇f) · ~u ∂x ∂y P0 " P0  P0 #

Definition 2 The gradient vector of f(x, y) at a point P0(x0,y0) is

∂f ∂f ∇f = ~i + ~j ∂x ∂y

Estimate Increment

The change of f when we move a small distance ds from P0 in the direction ~u.

∆f ≈ df = D~uf|P0 · ds =(∇f|P0 · ~u)ds

gradient, level curve, tangent line, normal line

Consider the multi variable function w = f(x, y). The level curve is f(x, y) = c and its vector form is ~r(t)= g(t)~i + h(t)~j, then

f(g(t), h(t)) = c

Differentiating both sides with respect to t

∇f · ~r′(t)=0 Lecture Notes of Section 11.5 Page 3 which implies gradient is normal to the level curve (~r′(t) denotes the tangent vector of the curve). Therefore, the equation for tangent line is

fx(x0,y0)(x − x0)+ fy(x0,y0)(y − y0)=0

the equation for normal line is

x = x0 + tfx(x0,y0)

 y = y0 + tf (x0,y0)  y  Definition 3 The tangent plane at the point P0(x0,y0, z0) on the level surface f(x, y, z)= c is the plane through P0 and normal to ∇f(P0). The equation form is

fx(P0)(x − x0)+ fy(P0)(y − y0)+ fz(P0)(z − z0)=0

HOMEWORK ASSIGNMENT

Page 923: 8, 11, 15, 21, 25, 29, 33, 37, 41, 47

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