Calc 3 Final

Front 1

Distance between points P1: (X1,Y1,Z1) and P2: (X2,Y2,Z2)

Back 1

sqrt( (X2-X1)^2 + (Y2-Y1)^2 +(Z2-Z1)^2)

Front 2

Formula for Dot Product

Back 2

a*b= a1b1+a2b2+a3b3

Front 3

Angle Between non zero vectors a and b

Back 3

Cos(Theta)= (a*b) / |a||b|

or

Sin(Theta) = |a x b| / (|a||b|)

Front 4

Two Vectors are Orthogonal if and only if

Back 4

a*b=0

Front 5

Scalar Projection of b onto a

Back 5

Compab= a*b / |a|

Front 6

Vector Projection of b onto a

Back 6

Projab= ((a*b)/ |a|) (a/ |a|)

or ((a*b) a) / |a|^2

Front 7

Two Non Zero Vectors a and b are parallel if and only if

Back 7

a x b = 0

Front 8

Area of parallelogram determined by a and b

Back 8

length of cross product

a x b

Front 9

How to find vector orthogonal to both a and b

Back 9

Cross product of a and b

a x b

Front 10

vector equation of L

Back 10

r=r0 + tv

Front 11

parametric equations of line L

Back 11

x= x0 + at

y= y0+ bt

z= z0 + ct

Front 12

symmetric equations of Line L

Back 12

(x-x0)/a = (y-y0)/b = (z-z0)/c

Front 13

Line segment from r0 to r1

Back 13

r(t)= (1-t)r0 + tr1

Front 14

vector equation of plane

Back 14

n*(r-r0)=0

or

n*r=n*r0

Front 15

scalar equation of plane through P0(X0,Y0,Z0) with normal vector n= <a,b,c>

Back 15

a(x-x0)+b(y-y0)+c(z-z0)=0

or

ax+by+cz+d=0

Front 16

formula for distance from point P1(X1,Y1,Z1) to plane ax+by+cz+d=0

Back 16

|ax1+by1+cz1+d|/ sqrt(a^2 + b^2 + c^2)

Front 17

formula for angle between planes

Back 17

(n1*n2)/(|n1||n2|)

Front 18

Formula for Ellipsoid

Back 18

(x^2 / a^2) + (y^2 / b^2) + (z^2 / c^2) =1

if a=b=c then it is sphere

Front 19

Formula for Cone

Back 19

(x^2 / a^2) + (y^2 / b^2) = (z^2 / c^2)

Front 20

Formula for Elliptic Parabeloid

Back 20

(x^2 / a^2) + (y^2 / b^2) = z/c

Front 21

Formula for Hyperbeloid of One Sheet

Back 21

(x^2 / a^2) + (y^2 / b^2) - (z^2 / c^2) =1

Front 22

Formula for Hyperbolic Parabeloid

Back 22

(x^2 / a^2) + (y^2 / b^2)= z/c

Front 23

Formula for Hyperboloid of Two Sheets

Back 23

(-x^2 / a^2) - (y^2 / b^2) + (z^2 / c^2) = 1

Front 24

Parametric Equations for Helix

Back 24

x= cost

y=sint

z= t

Front 25

Arc Length Formula 3-Dimensions

Back 25

Integral from A to B of

sqrt[ (f'(t))^2 + (g'(t))^2 + (h'(t))^2 ]

Front 26

Curvature of Curve

Back 26

K(t)= |r'(t) x r"(t)| / |r'(t)|^3

Front 27

Unit Tangent Vector

Back 27

T(t) = r'(t) / |r'(t)|

Front 28

Unit Normal Vector

Back 28

N(t)= T'(t) / |T'(t)|

Front 29

Binormal Vector

Back 29

B(t)= T(t) x N(t)

Front 30

Formula for speed

Back 30

ds/dt = |v(t)|= |r'(t)|

Front 31

Acceleration

Back 31

a(t)= v'(t)= r"(t)

Front 32

Velocity

Back 32

v(t)= r'(t)

Front 33

Parametric Equations of Trajectory

Back 33

x= (V0 cos(theta)) t

y= (V0 sin(theta)) t - (1/2)gt^2

Front 34

Tangential Component of Acceleration

Back 34

AT= ( r'(t)*r"(t) ) / |r'(t)|

Front 35

Normal Component of Acceleration

Back 35

AN = |r'(t) x r"(t)| / |r'(t)|

Front 36

Equation of tangent plane to surface z= f(x,y) at point P(X0,Y0,Z0)

Back 36

Z-Z0= Fx(Xo,Yo)(X-Xo)+ Fy(Xo,Yo)(Y-Yo)

Front 37

Linearization of f at (a,b) and the approximation

Back 37

f(x,y)= f(a,b) + fx(a,b)(x-a) + fy(a,b)(y-b)

Front 38

Linearization of f at (a,b,c) and the approximation

Back 38

f(x,y,z)= f(a,b,c) + fx(a,b,c)(x-a) + fy(a,b,c)(y-b) + fz(a,b,c)(z-c)

Front 39

Chain Rule for dz/dt

Back 39

dz/dt= (dz/dx)(dx/dt)+(dz/dy)(dy/dt)

Front 40

Chain rule for dz/ds

Back 40

dz/ds= (dz/dx)/(dx/ds)+ (dz/dy)(dy/ds)

Front 41

Implicit Differentiation

Back 41

dy/dx = - (df/dx)/(df/dy) = -Fx/Fy

Front 42

Directional derivative of f(x,y) in direction of u=<a,b>

Back 42

Du f(x,y)= Fx(x,y) a + Fy(x,y) b

make sure u=<a,b> is actually < a/|u|, b/|u|>

Front 43

Tangent Plane To Level Surface

Back 43

Fx(Xo,Yo, Zo)(X-Xo) + Fy(Xo,Yo,Zo)(Y-Yo)+ Fz(Xo,Yo,Zo)(Z-Zo)=0

Front 44

symmetric equations of tangent planes to level surfaces

Back 44

(X-Xo)/ (Fx(Xo,Yo,Zo))=

(Y-Yo)/ (Fy(Xo,Yo,Zo))=

(Z-Zo)/ (Fz(Xo,Yo,Zo))

Front 45

2nd Derivative Test

Back 45

D= D(a,b)= Fxx(a,b)Fyy(a,b) - [Fxy(a,b)]^2

a) If D>0, and Fxx(a,b)>0, then f(a,b) is local min

b) If D>0, and Fxx(a,b)<0, then f(a,b) is local max

c) If D<0, then f(a,b) is saddle point

Front 46

Moment about x-axis lamina

Back 46

Mx= double integral of y p(x,y)dA

Front 47

Moment about y-axis lamina

Back 47

My= double integral of x p(x,y)dA

Front 48

mass of lamina

Back 48

m= double integral of p(x,y)dA

Front 49

X coordinate of center of mass of lamina (2 dimensions)

Back 49

My/m = 1/m * double integral of x p(x,y)dA

Front 50

Y coordinate of center of mass of lamina (2 Dimensions)

Back 50

Mx/m = 1/m * double integral of y p(x,y)dA

Front 51

Moments of Inertia (2 dimensions)

Back 51

X axis: Ix= double integral of y^2 p(x,y) dA

Y axis : Iy= double integral of x^2 p(x,y) dA

Origin: Io= double integral of (x^2 +y^2)p(x,y)dA

Front 52

Mass of object (Triple Integral)

Back 52

m= triple integral of p(x,y,z) dV

Front 53

moment about yz plane

Back 53

Myz= triple integral of xp(x,y,z)

Front 54

moment about xy plane

Back 54

Mxy: triple integral of zp(x,y,z)

Front 55

Triple Integrals in Cylindrical Coordinates

Back 55

x=rcos(theta)

y=rsin(theta)

r^2 = x^2 + y^2

z=z

tan(theta)= y/x

dz= rdzdrd(theta)

Front 56

Triple Integrals in Spherical coordinates

Back 56

x=psin(phi)sin(theta)

y=psin(phi)cos(theta)

z=pcos(phi)

p^2 = x^2 +y^2 +z^2

dV= p^2 sin(phi) dp d(theta) d(phi)

Front 57

Formula For Line Integral

Back 57

Integral from a to b of

f(x(t),y(t)) * sqrt( (dx/dt)^2 + (dy/dt)^2 )

Front 58

Formula for Line Integral with respect to x

Back 58

Integral from a to b of

f(x(t),y(t)) x'(t) dt

Front 59

Formula for Line Integral with respect to y

Back 59

Integral from a to b of

f(x(t),y(t)) y'(t) dt

Front 60

Line Integral of F along C or F*dr

Back 60

Integral of F *dr

is integral from a to b of

F(r(t))* r'(t)

Front 61

Integral of Gradient Vector * dr

(Fundamental Theorem for Line Integrals)

Back 61

f(r(b)- f(r(a))

Front 62

F is conservative if

Back 62

DP/DY = DQ/DX

Front 63

Greens Theorem

Back 63

Double Integral of

(DQ/DX - DP/DY) dA

Front 64

Curl F

Back 64

(DR/DY - DQ/DZ) i + (DP/DZ - DR/DX) j + (DQ/DX -DP/DY) k

or

gradient X F

Front 65

Div F

Back 65

DP/DX + DQ/DY +DR/DZ

or

Gradient * F

Front 66

Surface Area

Back 66

Double Integral of

Sqrt( 1 + (DZ/DX)^2 + (DZ/DY)^2 ) dA

Front 67

Parametric Surface Area

Back 67

A(s)= double integral of |ru X rv|

Front 68

Surface Integral of F over S

Back 68

Double integral of f(x,y,z) ds

or

double integral of f(r(u,v)) |ru x rv|dA

Front 69

Integral of F * dr is equal to

Back 69

double integral of curl F * dS

Front 70

1

Back 70

1