Students starting out in the study of College Physics need to embrace the concept of mathematical modeling readily. Throughout most of your Physics course, mathematical modeling means no more than using algebra, geometry, trigonometry and calculus to represent physical phenomena. It is important to keep in mind that in physics, as distinct from pure mathematics, each graph and each mathematical expression tells a story about the real world we live in. The purpose I introduced early transcendentals in my tutorials is to lay the foundations to you for a smooth transition from Calculus to Ordinary First-Order Differential Equations.

Ignoring air resistance, if you fire a bullet 500 m/s horizontally from the top of a cliff that is 50 m above a long lake, and simultaneously drop a bullet off the cliff from the same height, they will both reach the ground at the same time. Of course, the dropped bullet (initial velocity is zero) will land on the ground vertically at 90° perpendicular to a flat ground.

Rotating just the i^ vector (x-axis) i' = i cos φ is incomplete. Technically, you have to rotate every vector in the vector space. Anyway, a picture is worth a thousand words!



Given that T is a linear transformation. To show that it rotates every vector in 2-dimensional vector space R² counterclockwise through the angle θ, let v = (x, y) be a vector in R². Using polar coordinates (which you probably have learned in Critical Tutorial #3: Complex Numbers polar form), you can write v and T(v ) as



respectively, where r = √(x² + y²) is the length (a.k.a. norm) of vector v and α (alpha) is the angle from the positive x-axis counterclockwise to the vector v. Now, applying the linear transformation T to produces



So the projected coordinates (x',y') of the point (x,y) after rotation are:



From the linear transformation above, R(θ) is known as the counterclockwise Rotation matrix:



If the direction of vector rotation is clockwise, then the Rotation matrix becomes:



This post has been edited by Critical_Fallacy: Jan 4 2014, 01:10 AM

Hi maximR,

You have seen how linear transformation works. Here is one real application of linear transformation. A robot can drive a certain distance or turn about a certain angle in its local coordinate system. For many applications such as Unmanned ground vehicle (UGV), Unmanned aerial vehicle (UAV), and Unmanned underwater vehicles (UUV), however, it is important to first establish a map (in an unknown environment) or to plan a path (in a known environment). These path points are usually specified in global or world coordinates.

Translating local robot coordinates to global world coordinates is a 2D transformation that requires a translation and a rotation, in order to match the two coordinate systems as shown in the Figure below.



Assume the Robot has the global position [Rx, Ry] and has global orientation ψ. It senses an Obstacle at local coordinates [Ox', Oy']. Then the global coordinates [Ox, Oy] can be calculated as follows:



For example, the marked position at the above Figure has local coordinates [0, 3]. The robot’s position is [5, 3] and its counterclockwise orientation is 30°. The global object position is therefore:



For computation purpose (perhaps kingkingyyk is interested ), the coordinate transformations can be greatly simplified by using homogeneous coordinates, where an arbitrary 2-D transformation sequences can be summarized in a single 3×3 matrix.

I would like to show you a quote from Galileo Galilei’s The Assayer (1623): “Philosophy is written in that great book which ever lies before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.”

Suggested route for learning Mathematical Physics:

(1) Matrix Algebra
(2) Vectors & Tensors
(3) Ordinary Differential Equations
(4) Partial Differential Equations

Effort spent in mastering these mathematical methods will be well rewarded!

To learn, you must understand the reason behind the answer to "why" in the first place.

Yesterday, have you learned the reason behind the answer (see Post #582) to your question in Post #563?

I ask because the way to answer your latest question is exact the same as in Post #582.

Can you sketch the Electric field around 'O'?

Remember the Superposition principle of potentials?



This post has been edited by Critical_Fallacy: Jan 4 2014, 08:06 PM

Just want to clarify with v1n0d. Isn't the application of four-color theorem shown on the cover of the book?

I'm pretty sure your textbook shows this formula for electric potential:



Find the Q of the Big droplet and radius r of the Big droplet.

To show maximR some useful applications of Differential Equations for Mathematical Modeling in Physical Sciences.

That probably means one thing, and indicate another thing.

The obvious thing is your understandings on the charge and the radius of the big droplet need to be corrected.

The second thing is to use mathematics to backup your reasons.

We cannot simply assume that the COMBINED charges of the big droplet reduces half to the individual charge of the small droplet.

Similarly, we also cannot simply assume that the radius of a larger VOLUME of spherical droplet is twice of the radius of the smaller VOLUME of spherical droplet.

I love Euler's solution of the Königsberg bridge problem!



This post has been edited by Critical_Fallacy: Jan 5 2014, 09:24 AM

danny88888 is right! Part (a) asks you to find the equations of all asymptotes.

Do you remember how to plot a Reciprocal Function: f(x) = 1/x in SPM Add Math?

Do you remember how to plot a Hyperbolic function in 1st Term Pure Math Chapter 5: Analytic Geometry?

Basically, to find the vertical and horizontal asymptotes, you have to find the LIMITS.



Next, you should find the turning points when the slope dy/dx = 0.



Using Cover-up method, you can find the partial fractions of the function.



Applying the superposition principle, you can sketch the graph!

+

+

+

=

Now, can you verify the asymptotes in Part (a) from this graph?

This post has been edited by Critical_Fallacy: Jan 5 2014, 09:39 AM

Have you found all the asymptotes?

If you want to factor , you can use CASIO fx-570 to aid you.

The built-in function "SOLVE" in the calculator uses Newton's method to find a real root, one at a time, if any.

From the function, it is not difficult to test a few points.



The results suggest that at least one real root lies between x: −2 & −1, and one real root lies between x: 0 & 1.

Picking the initial point x= −1, CASIO returns a real root x = −1.22074.

Similarly, picking the initial point x= 1, CASIO returns a real root x = 0.724492.



Performing the long division gives an irreducible quadratic expression:



Therefore, the factors are:

(1) Don't confuse Newton's Numerical Method (a.k.a. Newton-Raphson Method) with Newton's Law (F = ma).

(2)

(3) By picking the starting point x = 1, Newton's Method computes the roots by iterations: .

(4) Newton's iteration terminates if , that is when x = 0.724492.

Follow the instructions below to compute the root(s) numerically:

Are you tackling economic optimization problems?

THEOREM ::
Let be a function in n variables that depends on a parameter a. For each value of a, let be a maximum or minimum point for . Then

They are mostly Pre-U Maths and Physics, though there are some underground scientists!

You are welcomed to do discuss about AUSMAT Math and Physics here.

I can handle Introductory Statistical Methods only.

- Basic Probability,
- Discrete Random Variables,
- Continuous Random Variables,
- Point Estimation of Parameters and Sampling Distributions,
- Simple Linear Regression and Correlation
- Statistical Quality Control

For Advanced Statistical Methods made easy, you can probably consult a good teacher like mumeichan!

- Statistical Intervals for a Single Sample
- Tests of Hypotheses
- Statistical Inference for Two Samples
- Multiple Linear Regression
- The Analysis of Variance (ANOVA)
- Design of Experiments with Several Factors

Isn't it more important that we invite mumeichan to start a brief class on Applied Statistical Methods with Excel: for SPM & Pre-U Students?

Like how to determine Sample Size, Mean, Standard Deviation, and Standard Error of the Mean using =COUNT, =AVERAGE, and =STDEV functions...

Advanced topic may include finding the Confidence Interval about the Mean using the TINV Function...