Partial Derivatives And Multivariate Integration

Front 1

Limit in R3

Back 1

For a function to be continuous at a particular point it must approach a specific value from all directions.

Front 2

Partial Derivative notation

Back 2

If written f sub "variables" then you do the differentiation in the order the variable appear from left to right.

If it is written ∂/∂x∂y you start at the bottom right and move backwards towards the left.

Front 3

Clairaut's Theorem

Back 3

If Fxy and Fyx are both continuous then Fxy(x,y)=Fyx(a,b)

Front 4

Equation of the tangent plane to the surface z=f(x,y) @ point (x0,y0,z0)

Back 4

z - z0= Fx(x0,y0) (x-x0) +Fy(x0.y0) (y-y0)

Front 5

Mean Value Theorem

Back 5

F(x,y)-F(a,b)=∂f/∂x(a,b) * (x-a) +∂f/∂y (a,b) * (y-b) +e1(x-a) +e2(y-b)

Note: e1 and e2 are error factors and go to zero as x,y go to zero.

Front 6

Differential

Back 6

If z=f(x,y)
∂z= Fx(x,y) + Fy(x,y) = ∂f/∂x dx +∂f/∂y dy

Front 7

The Chain rule

Back 7

if q=f(a,b,x,y), x=f(t), y=g(t), a=(s,t), b=p(s)

∂q/∂t= (∂f/∂x)(dx/dt) +(∂f/∂y) (dy/dt) + (∂f/∂a) (da/dt)

∂q/∂s= (∂f/∂a)(da/ds) + (∂f/∂b)(db/ds)

Front 8

Variable Types for Chain rule

Back 8

Let z=f(x,y), x=g(s,t), y=h(q,r)

z is the dependent variable
x,y are called intermediate variables
q,r,s,t are independent variables.

Front 9

Directional Derivative

Back 9

∇f ∙ u (Where u is a unit vector in any direction)

The result is always a scalar.

Front 10

Maximum or Minimum rate of change in 3 variables

Back 10

It is |∇f(x,y,z)|

A Max is when u is in the same direction as the gradient

A min is when it is in the opposite direction as the gradient vector.

Front 11

Finding minima and maxima

Back 11

Take first order partial derivatives with respect to all variables and set them all equal to zero.

After that make a list of critical points and use that along with the extreme points of the Domain and Range to text in the original equation and see what the max and min values are.

Front 12

The "D" test

Back 12

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

a.) D > o and Fxx(a,b) >0 it is a min
b.) D > 0 and Fxx < 0 it is a max
c.) D< 0 Saddle point
d.) D=0, inconclusive.

Front 13

Fubini's Theorem

Back 13

If our function is continuous along the entire rectangular domain the order of integration doesn't matter.

Front 14

Type I Region

Back 14

A region is considered type I if we integrate "y" first and y is described as a function of x.

This means the first way we integrate will cause us to draw vertical lines if we are going from lower to upper bounds.

Front 15

Type II Region

Back 15

This means that we integrate with respect to x first. We will have x expressed as functions of y and when we first draw our lines to find the shape it will be horizontal lines.

Front 16

Polar Basics

Back 16

x=r cos(Θ)
y= r sin(Θ)
r^2=x^2 +y^2

DO NOT FORGET THE EXTRA R WHEN CHANGING TO POLAR

Front 17

∫sec^2(x)

Back 17

tan (x)

Front 18

∫csc^2(x)

Back 18

-cot(x)

Front 19

∫sec(x)tan(x)

Back 19

sec(x)

Front 20

∫tan(x)

Back 20

ln |sec(x)|

Front 21

∫sin^2(x)

Back 21

1/2x -1/4 sin(2x)

Front 22

∫cos^2(x)

Back 22

1/2x +1/4sin(2x)

Front 23

∫tan^2(x)

Back 23

tan(x) -1

Front 24

∫x sin(x)

Back 24

sin(x) - x cos(x)

Front 25

∫x cos(x)

Back 25

cos(x) +x sin(x)

Front 26

sin^2(x) +cos^2(x)

Back 26

1

Front 27

cos(2x)

Back 27

cos^2 (x) -sin^2(x)

Front 28

sin(2x)

Back 28

2sin(x)cox(x)

Front 29

sin(x)

Back 29

±√(1-cos^2(x))

Front 30

∂ sec(x)

Back 30

sec(x)tan(x)

Front 31

∂ tan(x)

Back 31

sec^2(x)

Front 32

∂ csc(x)

Back 32

-csc(x)cot(x)

Front 33

∂ cot(x)

Back 33

-csc^2(x)