Math & Physics Review MAR 555 – Intro PO Annie Sawabini

TA Contact Info  Annie Sawabini Office: SMAST, Room 110 Office: SMAST, Room 110 (508) Cell: (508) **be considerate** Cell: (508) **be considerate** Office Hours: Office Hours: Dartmouth: Tuesday and Thursday (drop by)Dartmouth: Tuesday and Thursday (drop by) Boston: Wednesday (by appointment)Boston: Wednesday (by appointment) Lowell/Amherst:Phone or Lowell/Amherst:Phone or

Topics  Coordinate Systems  Vectors Notation Notation Dot & Cross Products Dot & Cross Products  Derivatives Review Review Partials Partials Del Operator Del Operator Gradient, Divergence, Curl Gradient, Divergence, Curl  Motion – laws and equations  Miscellaneous

Coordinate System  Right hand coordinate system x y z [East] [North] [Up] Position u v w [eastward current] [northward current] [upward] Velocity 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 Notation  Scalars Magnitude only Magnitude only ex. Temperature or Pressure ex. Temperature or Pressure  Vectors Magnitude and Direction Magnitude and Direction ex. Displacement = distance (scalar) plus direction ex. Displacement = distance (scalar) plus direction a a a b b aa c a + b = c a + b = c a

Vector into Scalar Components  Resolving Vectors into Scalar Components on a 2D coordinate system x y a axax ayay Ø a x = a cos øa y = a sin ø sin = opposite hypotenuse cos = adjacent hypotenuse tan = opposite adjacent

Vector Operations  The dot product (aka. the scalar product) Two vectors produce a scalar Two vectors produce a scalar a b = a b cos ø  The cross project (aka. the vector product) Two vectors produce a vector that is orthogonal to both initial vectors Two vectors produce a vector that is orthogonal to both initial vectors a x b = a b sin ø a b

Derivatives  Derivative = the instantaneous rate of change of a function dy the change in y dxwith respect to x where y = f(x)  Also written as f´(x)

Derivatives  Example:  Remember why?

Derivatives  Power rule: f(x) = x a, for some real number a; f(x) = x a, for some real number a; f´(x) = ax a−1 f´(x) = ax a−1  Chain rule: f(x) = h(g(x)), then f(x) = h(g(x)), then f´(x) = h'(g(x))* g'(x) f´(x) = h'(g(x))* g'(x)  Product rule: (fg)´ = f´g + fg´ for all functions f and g (fg)´ = f´g + fg´ for all functions f and g  Constant rule: The derivative of any constant c is zero The derivative of any constant c is zero For c*f(x), c* f´(x) is the derivative For c*f(x), c* f´(x) is the derivative

Derivatives

Partial Derivatives  Partial derivative – a derivative taken with respect to one of the variables in a function while the others variables are held constant  Written:

Partial Derivatives  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 Written: Written: Note: i, j, and k are BOLD, indicating vectors. These are referred to as unit vectors with a magnitude of 1 in the x, y and z directions. Used as follows: Note: i, j, and k are BOLD, indicating vectors. These are referred to as unit vectors with a magnitude of 1 in the x, y and z directions. Used as follows: a = a x i + a y j + a z k

Gradient  Gradient – represents the direction of fastest increase of the scalar function the gradient of a scalar is a vector the 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 x direction anytime  T Example: temperature is said to have a gradient in the x direction anytime  T  x  x = 0

Divergence  Divergence - represents a vector field's tendency to originate from or converge upon a given point. Remember: the dot product of two vectors (F and  ) produces a scalar Remember: the dot product of two vectors (F and  ) produces a scalar Where F = F 1 i + F 2 j + F 3 k Where F = F 1 i + F 2 j + F 3 k

Curl  Curl: represents a vector field's tendency to rotate about a point Remember: the cross product of two vectors (F and  ) produces a vector Remember: the cross product of two vectors (F and  ) produces a vector For F = [Fx, Fy, Fz]: For F = [Fx, Fy, Fz]:

Newton’s Laws of Motion  First Law Objects in motion tend to stay in motion, objects at rest tend to stay at rest unless acted upon by an outside force Objects in motion tend to stay in motion, objects at rest tend to stay at rest unless acted upon by an outside force  Second law The rate of change of the momentum of a body is directly proportional to the net force acting on it, and the direction of the change in momentum takes place in the direction of the net force. The rate of change of the momentum of a body is directly proportional to the net force acting on it, and the direction of the change in momentum takes place in the direction of the net force.  Third law To every action there is an equal but opposite reaction To every action there is an equal but opposite reaction

Equations of Motion  Speed rate of motion (scalar) rate of motion (scalar)  Velocity = distance / time speed plus a direction (vector) speed plus a direction (vector)  Acceleration the rate of change of velocity over time the rate of change of velocity over time a = dv / dt average acceleration average acceleration a = (v f – v i ) / t  Force mass * acceleration mass * acceleration F = m*a F = m*a

Free body diagrams  Use to define all the forces acting on a body  Don’t forget to define your axes

Homework Expectations  Neat  Legible  Correct units always show always show  Show all work  Hand in on time

Questions  Questions?  Thanks to: Miles Sundermeyer and Jim Bisagni – presentation adapted from their lecture notes Miles Sundermeyer and Jim Bisagni – presentation adapted from their lecture notes Wikipedia – for pictures and equations ( Wikipedia – for pictures and equations (