1.2 Differential Calculus

1.2.1 The Gradient

y

zyx ˆˆˆz

T

y

T

x

TT

T(x,y)=const.

x

gradient:

cos

ˆˆˆ

ll

zyxl

dTdTdT

dzdydxd

dzdz

Tdy

dy

Tdx

dx

TdT

ld

T

points in the direction of maximum increase of T.It is perpendicular to the surface T=const.gives the slope (rate of increase) along this direction.T

T

θ

1.2.3 The “del” Operator

zyx

zyx ˆˆˆ

The del is similar to a vector, but it is an operator.It acts on (takes the partial derivatives of)everything to its right.

Example: Multiplication by a scalar-the gradient.),,( zyxf

The order of the factors does matter!

1.2.4 The Divergence

z

v

y

v

x

vzyx zyx

),,(v

divergence of a vector field

no divergence

large divergence

3 r

large divergence

z

z

2

ˆ2

v

zv

F(x,y) = (2*x,-y). F(x,y) = (2*x,-y)-(6,-1).

1.2.5 The Curl

Curl of the a vector field

)(ˆ)(ˆ)(ˆ

ˆˆˆ

),,(y

v

x

v

x

v

z

v

z

v

y

v

vvv

zyxzyx xyzxyz

zyx

zyx

zyx

v

Keep track of the order in evaluating the determinant!

G(x,y) - G(x0,y0)

(-y-3,x-3) (-y+3,x+3)

(x-y,x+y).

Turbulent motion of air around a vibrating cylinder in a wind tunnel.

1.2.6 Product Rules

• Apply del to all factors.• Keep track of the type of multiplication (dot

vs. cross, how connected).• Arrange in standard form (gradient, curl,

divergence).• Most important products are listed in the

book.• 0 )(3 rAArr

1.2.7 Second derivatives

component. each toApply field.a vector of ian Laplac

0)()(

ian Laplac)()(

:vector

22

2

2

2

2

2

22

v

TT

z

T

y

T

x

TTTT

T

)( :scalar vv

vvv

v

v

2

)()(

0)(

:vector