what is the Diagonal Rule?
445808
The diagonal rule is a method for determining the order in which electrons fill sublevels of energy levels of atoms when doing electron configurations.
706596
You'd have to use the chain rule for the square root, then the quotient rule for what's inside.\n\nf ' (x) = (1/2) [(2x³ - 7) / (4x + 1)]^(-1/2) times the derivative of the radicand.\nFor that, the quotient rule gives:\n[6x²(4x + 1) - 4(2x³ - 7)] / (4x + 1)²\n\nThere's most likely some simplifying to be done, but that should get you on the right path.\n\nGood luck!
746158
x^2 - x + some unknown value C\n\n2x^n+1 / n+1 is the rule in this case.  If you have a point, you can then plug it into the resulting equation to figure out "C"
419013
This is a cute one. Suppose A is the linear operator corresponding to the matrix A. \nThe equation implies A^2 - A = 0 which implies (A - I)A = 0 where I is the identity of the image of A. A could be the 0 operator; you have not said otherwise. It doesn't matter, because 0 is trivially diagonalizable. Suppose A is not the 0 operator (You can find an easy example in operators over R^2). Then A - I = 0 which implies A is the identity for some basis of its image which is the same as saying the matrix of A is diagonalizable.\nNote that the above is an equation of operators, and as was said, the I is not necessarily the identity of the original vector space. For example, a projection onto the x-axis also satisfies the above conditions and is diagonalizable. The identity in that case is over R.\nThis is cute because it has the analogue x^2 = x which has the only solutions 0 (the 0 operator) and 1 (the identity over R) over the reals. Over higher dimensions, this simple algebraic fact tells us a lot more.
588642
a^2 + b^2 = c^2\n\nthis assumes right angles, a and b are the length and width, c is the diagonal
894092
Here is a graph of .9 which is close to what you have: http://upload.wikimedia.org/wikipedia/en...\nThe way to read this is the diagonal is all r=1 and the rest of the relations are in the opposite positions. The .9 is in the first column... and the second box down. \n\n\nHere is a general graph that explains the relationships:\nhttp://hosting.soonet.ca/eliris/remotese...\n\nJust remember that just because it has a correlation of r=0 doesn't mean there isn't some correlation: http://www.uwsp.edu/psych/stat/7/i30.gif... ...\n\nThese points are obviously related and we can come up with a model to predict the next one.
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Well, I think I found something; it says that the rule is part of the Newton-Cotes formulas (Isaac Newton and Roger Cotes).\n\nThis is the page where I found the info:\nhttp://www.absoluteastronomy.com/ref/trapezium_rule
504740
After several attempts, it seems that it is rather impossible.  If you start in the bottom corner, you can get all but 3 squares rather easily.  If you're careful, you can have 1 remaining if you start diagonal to the corner.\n\nWhat are you trying to do this for?  Is it a website?  I'd love to try it again.
192120
Use the chain rule. Let f(y)=y^(1/2) and g(x)=x-2. Then your expression is just f(g(x)), and the chain rule states that D(f(g(x))) = Df(g(x))*Dg(x) = [(1/2)*(g(x))^(-1/2)] * [g'(x)] = [(1/2)*(x-2)^(-1/2)] * [1].\nI suggest doing several more of these questions until you no longer need to explicitly write out f and g, but can do so in your head.
475802
Basically, Cramer's Rule lets you solve for a variable in a system of equations via matrix operations without solving the whole matrix.  The process requires some knowledge of finding determinants.  The site below walks through an example of how this works.
409096
Use the line tool.  You can also use the rectangle tool and then transform the rectangle by hitting CTRL-T.  Move your cursor just outside the corner bounding box handle until it turns into the rotate cursor.  Rotate at will to create diagonals.
826187
I am guessing you want the vector projection onto one of the 6 sides rather than one of the 12 edges.\n\nThe diagonal vector is (1, 1, 1).  Projecting it onto any side simply removes the coordinate in the direction of projection.  Projecting it onto the side in the xy plane gives (1, 1, 0). \n\nIf you did mean an edge (the line where 2 sides meet), then the projection is always two 0s and a 1.  The one is in the direction of the edge.
903414
Because the diagonal creates two right triangles you can use the Pythagorean theorem to find the length of the other side:\n\n5^2 +x^2 = 9^2\n25 + x^2 = 81\nx^2 = 56\nsqrt of 56 = 7.48\n\nNow that you have the length of the other side you can use the formula    area = length x width:\n\n5 x 7.48 = 37.4 feet
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I don't know what is analatycal proof, but I'll solve it my own way. I hope it helps.\nA square is a rhombus with a right angle. The diagonals are equal and bisect each other.\nTo prove:The diagonals of the square are perpendicular to each other.\nTake a square ABCD\nDiagonals AC and BD intersect at O.\nAB = BC = CD = AD\nAC = BD\nAO = OC = BO = OD\nSo, all the triangles formed by the diagonals intersecting are congruent.\nSo,\nAngle AOB = Angle BOC = Angle COD = Angle DOA\nBut, \nAngle AOB + Angle BOC + Angle COD + Angle DOA = 360 degrees\n4(Angle AOB) = 360\nAngle AOB = 90\nHence proved
599592
At first I thought you were wanting the orthogonal matrices, but your example shows you are not. Orthogonal matrices are those whose inverse is their transpose. Yours are ones where the elements of the transpose are the inverse of the elements of the original.\n\nCertainly such a matrix has to have +-1 on the diagonal. Your matrix is singluar and any 2x2 matrix with your property and all 1 on the diagonal will be also. This means that one eigenvalue is 0 and the other is 2.\n\nI don't see anything special for 3x3 offhand, but I will look further.
451843
If you have correlation matrix A, then A(i,j) is the covariance between i and j, I think.  The matrix should be symmetric across its main diagonal.
352749
By the chain rule:\n[e^(3x^(2)-7x)]*(6x-7)