Calculus 1D With Raj, Judy & Robert

 Hyperbolic & Inverse  Contour Maps  Vectors  Curvaturez, Normal, Tangential  Parameterization  Coordinate Systems  Taylor Expanzion  Approximation  Projectile Motion  Keplers Laws of planetary motion  Vector Fields  Conservative  Line Integrals  Works  Curl & Divergence  Greens Theorem  Stokes Theorem  Surface Integration  Divergence Theorem  Maxwell’s Equations

 Construction of bridges  Hanging Cables or chains  Secondary Mirrors in Telescopes  Planetary Orbits  Field Deflection

 Restate a function in order to simplify its integration or derivation  t is often used, but it is just a variable name  This process simplifies integration of line integrals Parabolic Cylindrical Spherical

 What are Taylor Series Used for?  Limit of a Taylor Polynomial  Uses multiple derivatives in order to find an estimation at a nearby point.  More terms = better approx.  Let f be a function with derivatives of order k for k=1,2,…,N in some interval containing a as an interior point.  For any integer n from 0 through N The taylor polynomial of order n generated by f as x=a is the polynomial…

 Estimate the value of e x at 0.05  What do we have?  a = 0  f(x)=e x  f’(x)=e x Start with the derivatives at that point

 Adding terms to the taylor expansion leads to greater convergence onto the function  How did you think your calculator worked?  See Freddies multiple variable discussion

Magnitude Direction Scalar Multiplication Scaled Vector V = 4V = Dot Product Scalar Value |V||U|Cos( ß) VU = (Vx*Ux)+(Vy*Uy)+(Vz*Uz) Cross Product Orthogonal Vector |V||U|Sin( ß) VXU = Common Arithmetic Operations

Unit Tangent Binormal (Will be a unit vector) Unit Normal direction

 Defined by a Vector Function, as a function of each component  Curl & Divergence  Flow Patterns  Gradient

 Smooth Check is a matrix of all variables and their partials  We are effectively equating our vector field to the gradient of our function  Path Independence  Convert to polar  No Singularities  Magnetism  Gravity  Work Done

 Smooth Check  Check the partial derivatives  Integrate One Component  Constant function of others  Partial with respect to another component  y in this example  Compare it to the y component and solve for g’(y,z)  Repeat these two steps for the remaining components  A conservative vector field has a constant in the very end, relating it to no other variables

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 Curl is the tendency to rotate  Divergence is the tendency to explode

 The integral of the derivative over a region R is equal to the value of the function at the boundary B.  Divergence Theorem  R = Volume  B = Surface  Curl/Stokes Theorem  R = Surface  B = Line Integral

 Integrate to see how a field acts upon a particle moving along a curve.  Calculating the work done by a  force that changes with time  over a curve that changes with time  Estimating wire weight, given a density function

 Green is a simplification of Stokes, for 2D  Simple Jordan Curve

 Flux aka Surface Integration  Area Correction

 Fluids into an area  Based on volume changes CategoriesCategories: Structure of the Earth | Obsolete scientific theoriesStructure of the EarthObsolete scientific theories

1. The orbit of every planet is an ellipse with the sun at a focus 2. A line joining a planet and the sun sweeps out equal areas during equal intervals of time 3. The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit

 Originally a radical claim, because the belief was that planets orbited in perfect circles.  Ellipse for inner planets has such low eccentricity, so they can be mistaken for circles  The orbit of every planet is an ellipse with the sun at a focus  (r, theta) are heliocentric polar coordinates, p is the semi-latus rectum, and E is the eccentricity

 Planets move faster the closer it is to the sun  In a certain interval of time, the planet will travel from A to B  In an equal interval of time, the planet will travel from C to D  The resulting "triangles" have the same area  Conservation of angular momentum A line joining a planet and the sun sweeps out equal areas during equal intervals of time

 Is a way to compare the distances traveled between planets and how fast two planets travel, given the difference between the linear distances from the sun.  Example: Say Planet R is 4 times as far from the sun as Planet B. So R must travel 4 times as far per orbit as B. R also travels at half the speed of B, so it will take R 8 times as long to complete an orbit as B.  3) "The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit."  P is the orbital period of the planet and a is the semimajor axis of the orbit  Formerly known as the harmonic law

 Implications  Time Dilation  Relativity of simultaneity  Composition of velocities  Lorentz Contraction  Inertia and Momentum Cassini Space Probe & Relativity © nasa.gov