Math & Physics Math & Physics ReviewReview

MAR 555 – Intro POMAR 555 – Intro PO

Annie SawabiniAnnie Sawabini

TA Contact InfoTA Contact Info

Annie SawabiniAnnie Sawabini Email: g_asawabini@umassd.eduEmail: g_asawabini@umassd.edu

Office: Office: SMAST, Room 110SMAST, Room 110(508) 910-6321(508) 910-6321

Cell: (508) 264-6946 **be considerate**Cell: (508) 264-6946 **be considerate**

Office Hours:Office Hours:• Dartmouth: Dartmouth: Tuesday and Thursday (drop by)Tuesday and Thursday (drop by)• Boston: Boston: Wednesday (by appointment)Wednesday (by appointment)• Lowell/Amherst:Lowell/Amherst:Phone or EmailPhone or Email

TopicsTopics

Coordinate Systems Coordinate Systems VectorsVectors

NotationNotation Dot & Cross ProductsDot & Cross Products

DerivativesDerivatives ReviewReview PartialsPartials Del Operator Del Operator Gradient, Divergence, CurlGradient, Divergence, Curl

Motion – laws and equationsMotion – laws and equations MiscellaneousMiscellaneous

Coordinate SystemCoordinate System

Right hand coordinate systemRight hand coordinate system

x

y

z

[East]

[North]

[Up]Position

u

v

w

[eastward current]

[northward current]

[upward]Veloci

ty

Note: Ocean currents are named for the direction they are traveling in (ex. a northward current flows in the positive y). This is opposite the convention used for winds (ex. a north wind blows air from the north towards the south).

Vector NotationVector Notation

ScalarsScalars Magnitude onlyMagnitude only ex. Temperature or Pressureex. Temperature or Pressure

VectorsVectors Magnitude and DirectionMagnitude and Direction ex. Displacement = distance (scalar) plus ex. Displacement = distance (scalar) plus

directiondirection

a

a

a

b

b

a a

ca + b = c

a+b = c

a

Vector into Scalar ComponentsVector into Scalar Components

Resolving Vectors into Scalar Resolving Vectors into Scalar Components on a 2D coordinate systemComponents on a 2D coordinate system

x

y

a

ax

ay

Ø

ax = a cos ø

ay = a sin ø

sin = opposite

hypotenuse

cos = adjacent

hypotenuse

tan = opposite

adjacent

Vector OperationsVector Operations

The dot product (aka. the scalar product)The dot product (aka. the scalar product) Two vectors produce a scalarTwo vectors produce a scalar

a a •• b = a b cos b = a b cos øø

The cross project (aka. the vector product)The cross project (aka. the vector product) Two vectors produce a vector that is Two vectors produce a vector that is

orthogonal to both initial vectorsorthogonal to both initial vectors

a a xx b = a b sin b = a b sin øøa

b

DerivativesDerivatives

Derivative = the instantaneous rate of Derivative = the instantaneous rate of change of a functionchange of a function

dydy the change in ythe change in y

dxdx with respect to xwith respect to x

where y = f(x)where y = f(x) Also written asAlso written as

ff´́(x)(x)

DerivativesDerivatives

Example:Example:

Remember why?Remember why?

DerivativesDerivatives

Power rule: Power rule: f(x) = xf(x) = xaa, for some real number a; , for some real number a; ff´́(x) = ax(x) = axa−1a−1

Chain rule: Chain rule: f(x) = h(g(x)), then f(x) = h(g(x)), then ff´́(x) = h'(g(x))* g'(x)(x) = h'(g(x))* g'(x)

Product rule: Product rule: (fg)(fg)´́ = f = f´́g + fgg + fg´́ for all functions f and g for all functions f and g

Constant rule: Constant rule: The derivative of any constant c is zeroThe derivative of any constant c is zero For c*f(x), c* fFor c*f(x), c* f´́(x) is the derivative(x) is the derivative

DerivativesDerivatives

Partial DerivativesPartial Derivatives

Partial derivative – a derivative taken with Partial derivative – a derivative taken with respect to one of the variables in a respect to one of the variables in a function while the others variables are held function while the others variables are held constantconstant

Written:Written:

Partial DerivativesPartial Derivatives

Example:Example: Volume of a cone:Volume of a cone:

• r = radiusr = radius• h = heighth = height

Partial with respect to r:Partial with respect to r:

Partial with respect to h:Partial with respect to h:

, The Del Operator, The Del Operator The Del operatorThe Del operator

Written:Written:

Note: Note: ii, , jj, and , and kk are BOLD, indicating vectors. These are BOLD, indicating vectors. These are referred to as unit vectors with a magnitude of 1 in are referred to as unit vectors with a magnitude of 1 in the x, y and z directions. Used as follows:the x, y and z directions. Used as follows:

a = ax i + ay j + az k

GradientGradient Gradient – represents the direction of fastest Gradient – represents the direction of fastest

increase of the scalar function increase of the scalar function the gradient of a scalar is a vectorthe gradient of a scalar is a vector applied to a scalar function f:applied to a scalar function f:

Example: temperature is said to have a gradient in the Example: temperature is said to have a gradient in the x direction anytime x direction anytime T T

xx= 0

DivergenceDivergence Divergence - represents a vector field's Divergence - represents a vector field's

tendency to originate from or converge upon a tendency to originate from or converge upon a given point. given point. Remember: the dot product of two vectors (F and Remember: the dot product of two vectors (F and ) )

produces a scalarproduces a scalar

Where F = FWhere F = F11 i + F i + F22 j + F j + F33 k k

CurlCurl Curl: represents a vector field's tendency to Curl: represents a vector field's tendency to

rotate about a pointrotate about a point Remember: the cross product of two vectors (F and Remember: the cross product of two vectors (F and

) produces a vector) produces a vector For F = [Fx, Fy, Fz]:For F = [Fx, Fy, Fz]:

Newton’s Laws of MotionNewton’s Laws of Motion

First LawFirst Law Objects in motion tend to stay in motion, objects at Objects in motion tend to stay in motion, objects at

rest tend to stay at rest unless acted upon by an rest tend to stay at rest unless acted upon by an outside forceoutside force

Second lawSecond law The rate of change of the momentum of a body is The rate of change of the momentum of a body is

directly proportional to the net force acting on it, and directly proportional to the net force acting on it, and the direction of the change in momentum takes place the direction of the change in momentum takes place in the direction of the net force.in the direction of the net force.

Third lawThird law To every action there is an equal but opposite To every action there is an equal but opposite

reactionreaction

Equations of MotionEquations of Motion SpeedSpeed

rate of motion (scalar)rate of motion (scalar) Velocity = distance / timeVelocity = distance / time

speed plus a direction (vector)speed plus a direction (vector) AccelerationAcceleration

the rate of change of velocity over timethe rate of change of velocity over time

a = dv / dta = dv / dt average accelerationaverage acceleration

a = (va = (vff – v – vii) / t) / t ForceForce

mass * accelerationmass * acceleration

F = m*aF = m*a

Free body diagramsFree body diagrams

Use to define all the forces acting on a Use to define all the forces acting on a bodybody

Don’t forget to define your axesDon’t forget to define your axes

Homework ExpectationsHomework Expectations

NeatNeat LegibleLegible Correct units Correct units

always showalways show Show all workShow all work Hand in on timeHand in on time

QuestionsQuestions

Questions?Questions?

Thanks to:Thanks to: Miles Sundermeyer and Jim Bisagni – Miles Sundermeyer and Jim Bisagni –

presentation adapted from their lecture notespresentation adapted from their lecture notes Wikipedia – for pictures and equations Wikipedia – for pictures and equations

(www.wikipedia.org)(www.wikipedia.org)