Math & Physics Review MAR 555 – Intro PO Created by Annie Sawabini

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 (e.g. a northerly current is moving water along the positive y-axis). This is opposite the convention used for wind. A north wind blows FROM the north, along the negative y-axis.

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 aa

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 (a.k.a. the scalar product) Two vectors dotted together produce a scalar Two vectors dotted together 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

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:

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 In absence of external forces a body will move at constant velocity or stay at rest (or either depending on reference frame) In absence of external forces a body will move at constant velocity or stay at rest (or either depending on reference frame)  Second law Observed from an inertial reference frame, the net force on a particle is equal to the rate of change of its momentum F = d(mv)/dt. Or more simply force equals mass times acceleration. Observed from an inertial reference frame, the net force on a particle is equal to the rate of change of its momentum F = d(mv)/dt. Or more simply force equals mass times acceleration.  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