There is Elementary Column Operations but its not in the syllabus. If I am not mistaken, it has the same operations as ERO except it applies to a column instead of a row

Found a good site to study Multivariable .

http://ocw.mit.edu/courses/mathematics/18-...ulus-fall-2007/

Whether Shanks simplifies Rows 1 & 2 or not, it doesn't really matter, because the ultimate objective is to normalize all diagonal elements. It does not change the fact that Shanks still can reduce the augmented matrix to row echelon form within the minimum 6 STEPS in Gauss Elimination of a 3-by-3 matrix. before performing the back-substitution.

Can you find the real root of this polynomial using only SPM Coordinate Geometry skills?





Tips: a straight line, slope, multiple 2-point lines, root occurs @ y = 0

This post has been edited by Critical_Fallacy: Dec 19 2013, 04:20 PM

Simplification and normalizing the diagonal elements each have their own uses. For numerical work, the diagonal normalization reduces the number of operations required to obtain the inverse. In programming, this is the better approach because it cuts down on computing time. However, if the matrix is littered with unknown variables (such as algebraic expressions in lieu of numbers), it's not always the best idea to introduce fractions. In the case of exam questions, more often than not a matrix with unknown variables in it is reducible to a simpler form just by playing around with the rows. My advice is examine the matrix first and then decide which method is better.

Hi RED-HAIR-SHANKS & crazywing26,

Please listen to v1n0d's advice because he is an experienced math teacher. Perhaps I should make myself clear. If a Row can be simplified, it means the terms share a common divisor. When the diagonal element is normalized, the entire row is divided by the value of the diagonal element.

In fact, Gauss Elimination procedure does not require the diagonal elements to be normalized. Some students are not comfortable dealing with fractions. As long as you can reduce the entries below the diagonal elements to zeroes, then you solve the linear system. To avoid normalization, you can find the least common multiple between the diagonal element and the entries below it, so that you can perform the elimination.

For example, because a11 = 3, and a21 = 2, and the least common multiple is 6, therefore ERO {R2' = 3*R2 - 2*R1} will reduce a21 =0, without dealing with fractions. The is a price to pay: Be extra careful when doing the arithmetic calculations for the TWO Rows!



This post has been edited by Critical_Fallacy: Dec 19 2013, 09:11 PM

Hi RED-HAIR-SHANKS,





The idea behind using Coordinate Geometry to find the real root, of a curve is to exploit the slope formula of a straight line.

Since you know the real root of f(x) lies at the interval [4, 5], you can pick any two points within the interval, such that the lower bound , and the upper bound .

From these two points, and , a straight line can be drawn, where it intercepts x-axis. The x value that intercepts x-axis is closer to the real root, .

This post has been edited by Critical_Fallacy: Dec 19 2013, 09:40 PM

Thank you for posting your questions and also your guides on how to find the root of the quintic function. Well, actually I had a hard time cracking my brain in order to solve it. And most of all, I've failed to apply my knowledge of SPM level of Coordinate Geometry to find the root in this question. I've tried to delve a bit into this question, but, by using a different approach in order to find the x-value that is approaching to the real root.

At earlier of your posts, you gave me a tips by saying that the root occurs at f(x)=0. But, if we substitute 0 into y, we will get , and I guess it can't be solved, or perhaps I wasn't able to write down the root. I guess this has something to do with the Galois Theory, due to the fact that it's a quintic function. However, I can approximate the root by using Newton-Raphson method at least of it. Here it is:


From my above workings, notice that three of the final values above is slightly similar, which is 4.19272. Hence, from here, we know that it's getting even more closer and closer to the real root. Correct me if I'm wrong.

This post has been edited by RED-HAIR-SHANKS: Dec 19 2013, 11:44 PM

Any tips to find the boundary in the subtopic change of variables in multiple integrals?.. the only problem for me is to find the boundary after I did jacobian

Ya, some of the questions are too tough. You can do all the exercise in three books and make the comparison yourself.

Gravitation is interesting. Torque is fine. Be worried about kinematics and dynamics. Although you've learnt all the concepts in SPM, STPM questions are much tougher. You can use your SPM level knowledge to do the first three chapters, all you need now is think better. Twist your brain harder.

Thermodynamics (kinetic theory of gases) is the most interesting for first term, at least to me.


Why delve so far into newton-raphson? In most cases, you can approximate the root by using trial and error. If the root involves decimal places, you can use either iteration method, newton-raphson or trapezium rule. However, for first term Maths, none of the calculus method is required. The roots are normally fractional or integral.

Differentiation (Newton's method) is not required although it can be very efficient. I wouldn't train you how to solve if it cannot be solved using only SPM Coordinate Geometry. Like I told you, pick 2 points within [4, 5], and with good judgment, I'd pick xL = 4.00 and xU = 4.25. To complete the info of points, find f(4.00) and f(4.25). Now you can find the slope (m) of the straight line between these two points.



With the slope and one of the points, you can find the equation of the straight line. To find the x-intercept of the straight line, just let y = 0. The x-intercept must be within the interval and is getting closer to the real root. Now, check if you understand the following algorithm (procedure).



It is convenient to tabulate your calculations using a table. I'll show the first 2 iterations and you can do the rest. In reality, sometimes you will encounter non-differentiable functions. So, you cannot use Newton's method to solve it. I discover this method independently when I was Form 4 (no one taught me Newton's method). My purpose is to show you how powerful SPM Add Maths can be. Of course, this requires Higher Order Thinking Skills (HOTS)!



This post has been edited by Critical_Fallacy: Dec 20 2013, 01:59 AM

Your answer is correct, but you used Newton's method. Perhaps this graphical representation of Coordinate Geometry will give you the better idea. As you can see, the x-intercept is getting closer to the true root of the curve. By the way, this SIMPLE root-finding technique is known as the False Position method.

can I ask how to solve this? =( Sorry I'm that bad at self learning .....


Thanks =D

For parts (a) and (b), you will need to either:
* sketch the graph of f(x) for some interval containing the value x=3 (say 0<x<6)
OR
* draw a table and compute values of f(x) as they approach from the negative and positive directions (compute f(x) for values of x=2.8,2.9,3.1,3.2 etc.)

The third part is easily completed by recalling that the limit exists at a point if the limits from the positive and negative direction are equal to each other, i.e. f(x) has a limit at x=3 if your results for (a) and (b) are the same.

P.S. Critical_Fallacy I tried using the TeX editor in your siggy, but it seems quite tedious. Do you upload each image to Imgur or am I doing it wrong?

Consider the definition of absolute function of |x − 3|:



It is not difficult to see that



This means that no matter how close x gets to 3, there will be both positive and negative x-values that yield f(x) = 1 or f(x) = −1. This is clearly illustrated on the graph of the function:



Because approaches a different number from the right side of 3 than it approaches from the left side, the limit does NOT exist.

It's good to see you back in this forum after you've gone out for a while . While we're on it, mind if I ask concerning the bold statement above as to why should I be dreaded with kinematics and dynamics? I don't mean to underestimate both of them, it's just that I can at the very least of it get a little gist or idea on what should I face and learn in the foreboding future of my STPM.

I don't think I've delve too far or deep into the Newton-Raphson method. It just so happens that I found out the usage of this method from my brother's book. And for your info, I'm quite new to all of these, so, I might take some time to get the hang of it.

Thanks for your guidance and solutions. I'll try to take my time to delve deeper into it. Oddly, for some unknown reason, I'm more incline towards Newton-Raphson method compare to iteration method in this case.

It's quite hard to tell why. Perhaps you should try to do the questions on the topics in the books, then you might understand why. Be prepared for an initial shock, and don't treat it with SPM thinking skills. Use the same concept, but twist your brain differently.

Maths is a very interesting subject. But try to focus on first term first, because there are way too much to study for first term and you wouldn't want to be bothered by the second or the third.

And I'll be away for quite a while. I've begun working and won't have time to go online often.

where the Jacobian matrix is given by



Do you mean finding a transformation from a region S in the uv-plane, to a region R in the xy-plane?

I was pleasantly surprised this morning when I got up to see the whole lot of you rallying behind getting TeX support for our forum. Thanks guys!

For the time being, we'll make do with the online service that wKkaY suggested. The link to it is in Critical_Fallacy's siggy.

On a separate note, Critical_Fallacy are you familiar with integration over matrices? I need to compute the Riemannian metric over the Fisher matrix but I'm totally lost.