partial derivatives and their application

8/12/2019 Partial derivatives and their application

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Math 2080 Week 10 Page 1 Gentry Publishing

Chapter 10 Partial derivatives and their application.

10.1 Partial Derivatives

10.2 Tangent Planes and slopes of surfaces.

10.3 Linear approximations and

the differential of F(x, y).

10.4 Linear Stability analysis of multivariatedynamical systems.


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The first partial derivative of z = F(x, y) withrespect to x is

F(x, y) =M M x

limh60

F ( x% h, y)& F ( x, y)h

provided the limit exists.

This partial derivative is also denoted by

zx M z M x

and

Fx(x, y) or, simply F x.


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The first partial derivative of F(x, y) withrespect to y is

F(x, y) =M M y

limh60

F ( x, y% h)& F ( x, y)h

provided the limit exists.

This partial derivative is also denoted by

, zy,M z M y

and

Fy(x, y) or simply F y.


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The four second partial derivativesof z = F(x, y) are :

The second partial derivative of F with respect to x:

M2/Mx2 F(x, y) = F(x, y) = F xx(x, y)M2 M x 2

/ Fx(x, y) = F(x, y) M M x

M M x

M M x


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The second partial derivative of F with respect to y:

M2

/My2

F(x, y) = F(x, y) = F yy(x, y)M2 M y 2

/ Fy(x, y) = F(x, y) M M y

M M y

M M y


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The (mixed) second partial derivative of F withrespect to x and then y:

M2/MyMx F(x, y) = F(x, y) = F xy(x, y)M2 M yM x

/ Fx(x, y) = F(x, y) M M y

M M y

M M x

The (mixed) second partial derivative of F withrespect to y and then x:

M2/MxMy F(x, y) = F(x, y) = F yx(x, y)M2 M xM y

/ Fy(x, y) = F(x, y) M

M x

M

M x

M

M y


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Problem Determine all first and second order partial derivatives of the given function.

(a) F(x, y) = x3

+ x2

y - y4

(b) F(x, y) = x sin(y)


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(c) F(x, y) = x ln(y2) - e yx


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If the partial derivatives F xy and F yx arecontinuous in a circular region about a point(x, y) then

Fxy(x, y) = F yx(x, y) .

EQUALITY OF MIXED PARTIALDERIVATIVES .


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These concepts extends to functions of more thantwo variables, e.g..

If G(x,y,z) = xy -yz2

then

Gx =

Gz =

Gzy =

Gzx =


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For a function G of x, y, and z, there are 24 mixed partial derivatives that differentiate twice withrespect to x, twice with respect to y and once with

respect to z:

Gxxyyz G xxyzy G xxzyy G xzxyy G zxxyyGxyxyz G xyxzy G xyzxy G xzyxy G zxyxyGyyxxz G yyxzx G yyzxx G yzyxx G zyyxxG

yxyxz G

yxyzx G

yxzyx G

yzxyx G

zyxyxGyxxyz G yxxzy G yxzxy G yzxxy G zyxxyGxyyxz G xyyzx G xyzyx G xzyyx G zxyyx

If G is smooth enough, all of these will have thesame value.

What does this mean?

Eg. if

G(x,y,z) = x sin(y) + xe-zy

- y x% ln( x% z 2)

x z & 2 x 2% cos( x/ z )

which way would you evaluate G zxyyx ?


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Planes in 3-dimensional space.

Linear FUNCTION EQUATIONS

2-Dimensional Line: y = mx + b

3-Dimensional Plane: z = Ax + By +C

more generally:

4-dimensions: Hyper-space

In N-dimensions, a linear surface is an N-1dimensional Hyper space .

N-dimensional vector:


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y

1

2

3

4

5

x1 2 3 4 5

POINT-SLOPE EQUATIONS

2-D Line: y = y 0 + m(x - x 0)

3-D Plane : z = z 0 + A(x - x 0) + B(y - y 0)


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y

1

2

3

4

5

x1 2 3 4 5

INTERCEPT EQUATIONS

2-D Line: xa

% yb

' 1


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x y

z

x y

z

X/4 + y/2 + z/3 = 1

3-D Plane: xa

% yb

% z c

' 1


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Planes parallel to two of the coordinate axes .

Each pair of coordinate axes determine a plane, e.g.,

the x-y plane, the y-z plane.

These planes are characterized by the fact that thethird coordinated is zero at each point in the plane.

The x-y plane is

The x-z plane is

The y-z plane is


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More generally, planes that are parallel to one of these two-axis planes are characterized by the

property that one of their coordinates is constant.

{ (x, y, z) | }

This plane is then parallel to the plane formed fromthe two non-constant coordinate axes.


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Parallel Lines

L1 : y = y 0 + m(x - x 0)

Slope: Point:

Two lines are parallel if _________________________________.

A line parallel to L 1 is

L2: y =

What if the slope is m = P/Q?

Then we could write the line L 1 as

(y - y 0) - P/Q (x - x 0) = 0

or

(x - x 0) (y - y 0) = 0


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y

-5

-4-3

-2

-1

1

2

3

4

5

x-5 -4 -3 -2 -1 1 2 3 4 5

y

x

grid axis -6 to 6


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y

-5

-4

-3

-2

-1

1

2

3

4

5

x-5 -4 -3 -2 -1 1 2 3 4 5

What is a vector?

What is the (position) vector v = (Q, P) ?

What is the relationship of the vector v = (Q, P)to the line L 1?


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(Parametr ic) Vector Equation of a L ine

L1 = { (x, y) | (x, y) = (x 0, y 0) + t(Q, P)

for t 0 (-4 , 4 ) }


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Perpendicular LinesTwo lines are perpendicular if

____________________________________

What is the slope of the line perpendicular tothe line L 1?

Slope of :L 1: m =

Slope of perpendicular line L 1-N:8

m =

Equation of perpendicular line:

Do you know another word that in mathematicsmeans perpendicular?

________________

The line L 1-N is ___________ to the line L 1 atthe point (x 0, y 0).


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Dot Products

The dot product or inner product of two

vectors is the sum of the product of their respective components:

Let V = (v 1, v 2) U = (u 1, u 2)

the dot product of V and U is

VCCCCU / v1u1 + v 2u2

If V = (2, -3) U = (4, 5) then VCCCCU = __________

In 3-dimensions,

if N = (2, 3, 5) and T = (-1, 7, 4)then

NCCCCT = ___________________


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U _

v_

O-

The dot-product of two vectors can also be expressedas

uCv = ||u|| ||v|| cos( )

where is the angle between the two vectors, and

|| u || is the length of the vector u:

(u 12 + u 2

2)

Notice that since

-1 # cos( ) # 1

we have

- ||u|| ||v|| # uCv # ||u|| ||v||


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U _

v_

O- B

A

C vP

Geometrically, the dot product is the product ofthe length of the vector u

and

the length of the projection of the vector v onto u

this is how much v goes in the direction of u

In the diagramat the right, thelength A is ||v||, andC is the length of the

projection of v ontou ,

This is the distancefrom the commonvertex to the pointPv which is given as

C = ||v|| cos( ) since cos( ) = C/A.


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What is a vector perpendicular to agiven vector v = (v 1, v 2) ?

N = ( , )

If two vectors are perpendicular what is their dot product?

If N is perpendicular to v then N C v =


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y

-5

-4

-3

-2-1

1

2

3

4

5

x-5 -4 -3 -2 -1 1 2 3 4 5

Returning to the line

L1 = { (x, y) | (x, y) = (x 0, y 0) + t(Q, P)

for t 0 (-4 , 4 ) }

What is the vector N = (P, -Q)?

What is the relationshipof the vector N = (P, -Q) to the line L

1?


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The Normal Equation of the line through the point (x 0, y 0)

perpendicular to the vector N = (P, -Q):

L1 = {(x, y) | [(x, y) - (x 0, y 0)] C (P, -Q) = 0}

or , without using the set notation

L1: (x - x

0 , y - y

0) C(P, -Q) = 0

or

L1: P(x - x 0) - Q(y - y 0) = 0


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Exercises. What is a point on the given line andwhat is a normal vector to the line?

(a) 2(x - 3) + 3(y + 2)= 0

Point: (x 0,y0) = Normal vector: N =

(b) y = 4x+ 5


(c) x/2 + y/3 = 1



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The Normal Equation of a Planethrough the point (x 0, y 0, z0)

perpendicular to the vector (n x, n y, n z):

P1 = {(x, y, z) |[(x, y, z) - (x 0, y 0, z0)] C (n x, n y, n z) = 0}

or , without the set notation

P1: [(x, y, z) - (x 0, y 0, z0)] C (n x, n y, n z) = 0

or

P1: n x(x - x 0) + n y(y - y 0) + n z(z - z 0) = 0

If we expand the last equation we get the morecommon equation for a plane

____ x + ___ y + ___ z = _____

or, if n z 0, solving for z

z = ____x + ____ y +


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What is a normal vector to the plane

(d) 2(x - 3) + 3(y + 2) - (z -5) = 0

N =

(e) z = 4x - 2y + 5

N =

(f) x/2 + y/3 + z/5 = 1

N =


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TANGENT Lines and SURFACESas lines or planes perpendicular to a normal vector:

2-D: Tangent line to y = f(x) at (x 0, y 0)is the line perpendicular to the Normal vector: (f N(x0), -1)

f N(x0)(x - x 0) + -1(y - y 0) = 0

y

1

2

3

4

5

x1 2 3 4 5


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3-D: Tangent Plane to z = F(x, y) at(x0, y 0, z 0) is the plane perpendicularto the normal vector

N = ( F x(x0, y 0), F y(x0, y 0), -1)

Fx(x0, y 0)(x - x 0) + F y(x0, y 0)(y - y 0) - 1(z - z 0) = 0

orz = z 0 + F x(x0, y 0)(x - x 0) + F y(x0, y 0)(y - y 0)


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TANGENT Lines and SURFACES

2-D: Tangent line to y = f(x) at (x 0, y 0)

y = L(x) = y

1

2

3

4

5

x1 2 3 4 5

3-D: Tangent Plane to z = F(x, y) at (x 0, y 0, z0)

z = P(x, y) =


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x

y

z

Line of intersection of two planes.

In 2-DThe intersection of 2 lines is __________

In 3-D.

The intersection of two planes is _____________


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Line in three-space as theintersection of two planes.

L = {(x, y, z) * z = z 0 + A 0 x + B 0y and

z = z 1 + A 1 x + B 1y }


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Line in three-spaceas the intersection of two planes.

If (x 0, y 0, z0) is a point on the line L which is theintersection of the planes

z = z 0 + A 0 x + B 0y and z = z 1 + A 1 x + B 1y

Then the line L is the set of points thatsimultaneously satisfies the point-slope equations

A0 (x - x 0) + B 0(y - y 0) + C 0( z - z 0) = 0 and

A1 (x - x 0) + B 1(y - y 0) + C 1( z - z 0) = 0

with C 1 = C 0 = _____

Example: What is the line given by x= y and y = z ?

What is the line given by y = 2 and z = 3

partial derivatives and their application

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