L06 Gradient and Directional Derivatives_annotated 09032012

8/2/2019 L06 Gradient and Directional Derivatives_annotated 09032012

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ENG 2091

Advanced Engineering Mathematics A

Dr Lau Ee Von

Lecture 6

Gradient and Directional

Derivatives


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Gradient of a Scalar Field/Function

Gradient of a scalar function , , isdenoted by

grador

grad = =

,

,

=

+

+

vector field/function


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Gradient of a Scalar Field/Function

ExampleFind the vector field (gradient) of , , = 2 + 3 + at P(2,1,3)

grad = =

,

,

= [4, 6, 2]

at P(2,1,3)= 8, 6, 6 = 8 + 6 + 6


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Directional Derivative

Directional derivative of a function , , ,

or

Definition: The rate of change of , , in anarbitrary direction, in space

grad or = vector field

Definition: The rate of change of (,,) in thex,y,z-coordinate axes direction


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Directional derivative of a function , , at apoint, P

=

= lim

()

P

0

Q

s

sis the arc length of line L|s|is the distance between P0and Qs>0if Qlies in the direction of s


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Directional derivative of a function , , at apoint, P0

=

= lim

()

P0

Q

s

sis the arc length of line LEquation of line L (parameterised by arclength, s)

= + + = +

L


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=

Equation of line L (parameterised by arclength, s)

= + + = +

() =

=

+

+

=

=

+

+

=

=

=


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=

=

=

= = 1

||

If the direction is givenas a unit vector,

If the direction is givenas a vector,


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Example

Find the directional derivative of , , = 2 + 3 + at P(2,1,3) in the direction of a= [1, 0, -2]

=

,

,

= [4, 6, 2]

() = 8, 6, 6

=

1

||

=1

1 + (2)1,0, 2 8,6,6

= 1.789

Directionalderivative of a

point is aScalar


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We know the definition of and

But what does it all mean?

=

= cos

Since represents a unit vector, therefore = 1 = cos

If =

, then = 0

Vector field is perpendicularto the point on

curve/surfacei.e. surface normal vector,

Directional derivative = 0, i.e.scalar function (level curve or

level surface)


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Confused?

In essence,

is constant (i.e. either levelcurves or level surfaces)

= , surface normal vector


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The directional derivative, (of a scalar field,) at a point attains a maximum value whentaken in the direction of the gradient of

(direction of the steepest ascent is in the samedirection of the vector field at that point)

We know the definition of and

But what does it all mean?

The directional derivative, (of a scalar field,

) at a point attains a minimum value when takenin the direction opposite to the gradient of (direction of the steepest ascent is in theopposite direction of the vector field at thatpoint)


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Level curves of hill height

Even more confused?


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Level curves of temperature away fromfire

Direction of steepest ascend/rate ofincrease

Direction of steepest descend/rate of

decrease

= +ve

= -ve


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Past year exam question: Q1,2008


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Quotient rule:

()

()

= ()

[ ]


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Summary

Gradient, of a scalar field =vector field

Directional derivative = rate ofchange of a scalar field at apoint Pin a specified direction

= , surface normal vectorfor level curves/surfaces

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L06 Gradient and Directional Derivatives_annotated 09032012