103-102 CALCULUS II
This presentation is the property of its rightful owner.
Sponsored Links
1 / 66

103-102 CALCULUS II 2 PowerPoint PPT Presentation


  • 95 Views
  • Uploaded on
  • Presentation posted in: General

CALCULUS II. Three dimensional space. Scalar Product and Vector Product. Lines and Planes in space. Vector-Valued Functions and Space curves. Arc Length and the Unit Tangent Vector. Functions of many variables. Partial Derivatives and Chain Rule. Directional Derivatives, . Gradient Vectors and Tange

Download Presentation

103-102 CALCULUS II 2

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


103 102 calculus ii 2

103-102 CALCULUS II 2

CALCULUS II 2

3 , , (Vectors) 3

3 , 3


103 102 calculus ii 2

CALCULUS II

Three dimensional space

Scalar Product and Vector Product

Lines and Planes in space

Vector-Valued Functions and Space curves

Arc Length and the Unit Tangent Vector

Functions of many variables

Partial Derivatives and Chain Rule

Directional Derivatives,

Gradient Vectors and Tangent Planes

Extreme values and Saddle Points


103 102 calculus ii 2

(2 )


103 102 calculus ii 2

()

Cartesian (Rectangle) Coordinate

(0,0,0)


103 102 calculus ii 2

(Quadrant)


103 102 calculus ii 2

(Octant)

3

8 (Oct = = 8)


103 102 calculus ii 2

(1st Octant)

(2nd Octant)

(3rd Octant)

(4th Octant)


103 102 calculus ii 2

(5th Octant)

(6th Octant)

(7th Octant)

(8th Octant)


103 102 calculus ii 2

xy (xy-plane)

xy

(x,y,0)


103 102 calculus ii 2

xz (xz-plane)

xz

(x,0,z)


103 102 calculus ii 2

yz (yz-plane)

yz

(0,y,z)


103 102 calculus ii 2

x (x-axis)

x

(x,0,0)


103 102 calculus ii 2

y (y-axis)

y

(0,y,0)


103 102 calculus ii 2

z (z-axis)

z

(0,0,z)


103 102 calculus ii 2

=


103 102 calculus ii 2

2


103 102 calculus ii 2

2


103 102 calculus ii 2

(2,3,-1) (4,-1,3)


103 102 calculus ii 2

(1,1, 3) (-3,-1,-1)


103 102 calculus ii 2

(0,0,0)

5


103 102 calculus ii 2

(0,0,0)

5


103 102 calculus ii 2

(x0, y0, z0)

r


103 102 calculus ii 2

(x0, y0, z0)

r


103 102 calculus ii 2

(-1,0,-4)

3


103 102 calculus ii 2

(-1,2,-3)

11


103 102 calculus ii 2


103 102 calculus ii 2


103 102 calculus ii 2


103 102 calculus ii 2

2

M

2 P1(x1, y1, z1) P2(x2, y2, z2)


103 102 calculus ii 2

P1(3,-2,0) P2(7, 4, 4)


103 102 calculus ii 2

(Vector)

!!!


103 102 calculus ii 2


103 102 calculus ii 2

(1,2,1)

(0,1,1)

(1,0,0)

(2,1,0)


103 102 calculus ii 2

0 0 0


103 102 calculus ii 2

1.

1.1 v+w = w+v

1.2 0+v = v+0 = v

2


103 102 calculus ii 2

2. (scalar) ()

2.1

2.1.1

2.1.2


103 102 calculus ii 2

2.2

2.2.1

2.2.2


103 102 calculus ii 2

u v 0

u v

u = v

scalar


103 102 calculus ii 2

v-w = v+(-w)

w

v

w

v+w

v

-w

-w

v+(-w)=v-w

v


103 102 calculus ii 2

v-v = v+(-v)=

v

-v


103 102 calculus ii 2

(Considering vector in component form)


103 102 calculus ii 2

i,j k

i 1

x

k

j

j 1

y

i

k 1

z


103 102 calculus ii 2

i,j k

v=v1i +v2j +v3k

v

v=<v1, v2, v3>


103 102 calculus ii 2

2

<a,b,c> = <-2, , >


103 102 calculus ii 2

v=<v1, v2, v3> w= <w1, w2, w3>

v+w= < v1+w1, v2+w2, v3+w3>

= (v1+w1)i+(v2+w2)j+(v3+w3)k

v-w= < v1-w1, v2-w2, v3-w3>

= (v1-w1)i+(v2-w2)j+(v3-w3)k

v= < v1, v2, v3>

= ( v1)i+( v2)j+( v3)k


103 102 calculus ii 2

v=<-2,1,0> w= <3,-4,-5>

v+w=

0.5v=

-2w=

2w-3v=


103 102 calculus ii 2

P1 P2

P1 P2


103 102 calculus ii 2


103 102 calculus ii 2

P1 P2

P1 P2

P2(x2, y2, z2)

P1(x1, y1, z1)


103 102 calculus ii 2

3


103 102 calculus ii 2

P1(0,-2,5) P2 (3,4,-1)

P1 (0,-2,5) P2(3,4,-1)


103 102 calculus ii 2

u,v w

k l


103 102 calculus ii 2


103 102 calculus ii 2

v=<v1, v2, v3> v

v


103 102 calculus ii 2

v=<v1, v2, v3> v

v


103 102 calculus ii 2

v= <3,-4,-5>

w= 3i+4j+5k

2v

-3w

0


103 102 calculus ii 2

v 0 1

v

(normalization)


103 102 calculus ii 2

1

v= <2,-2,1>


  • Login