Math Is Fun Forum

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So do you mean Differential Calculus consists of Differential Equations and Partial Differential Equations?

you can plot "y=x*x*sqrt(36-x*x)" for xMin=0 to xMax=6 and parameters none at this site:

http://www.wessa.net/math.wasp

2√6≈4.8989
the theorem is famous in the field of inequalities, which is neglected by standard math education.

What's the difference between Caculus formulas and Differencial Calculus formulas?

Recently I read some books on business. One of them say it was Apple's fault to persue a flawless system at the cost of openess and software variety.

Products of Vectors and Matrices

Scalar Product

A vector or a matrix can be multiplied by a scalar k entirely.
For Example

Vector Product

two vectors containing same amount of entries can be multiplied. some people call this "dot product"

or equivalently, using matrix rule of product expression:

notice the second vector is placed verticle in matrix rule of product expression.

We can think the product as each entry of the former vector(a,b and c) , is scalar multiplied by corresponding entry of the latter vector, and then the 3 product ad, be and cf are added up and give the final result. So does the reverse.(this concept will be applied below next *)

Extension to Matrix Vector Product

We can add another vector [a[sub]2[/sub] b[sub]2[/sub] c[sub]2[/sub]]under the vector [a b c] and let it do the SAME multiplication to [d e f], and do the SAME summation and the result is placed under previous one for vector[a[sub]2[/sub] b[sub]2[/sub] c[sub]2[/sub]] has been placed under[a b c]:

Recall the concept of scaler product-sum analyze, we will notice both a and a[sub]2[/sub] are multiplied by scalar d, both b and b[sub]2[/sub] are multiplied by e, as well as both c and c[sub]2[/sub] are multiplied by f, and then corresponding product are added. If we define

then the matrix product can be expressed as

where col stands for column, A is now a matrix and no longer a vector .

we can add many rows
[a[sub]3[/sub] b[sub]3[/sub] c[sub]3[/sub]]... [a[sub]m[/sub] b[sub]m[/sub] c[sub]m[/sub]] to the matrix, but

still holds.
or

This is called a matrix multiplied by a vector on the right is equivalent to get its columns(also vectors) combined by entries of the right vector.It's another way to perceive matrice product.

Similarly, we have this formula about row combination*

Extension to Matrice Product
what if  a matrix multiplied by a matrix? we can either seperate right matrix(B) into columns of scalars and get a Row of Column combinations or seperate left matrix(A) into rows of scalars and get a Column of Row combinations.

MIF you are so creative! How did you do it?

to compute the area enclosed by an ellipse.

toss a dime, the chance of getting head up is 1/2. toss two dimes, the chance of getting two heads up is 1/2 (1/2) =1/4 , the 1/2 in the bracket is the chance of getting head up for the second dime, giving the first dime heads up. it's the same as the chance of getting head up without the first dime. This is called probalisticly independent.

the expert win = he beats all  novices = all novices fail to win him = 1st novice fails, same time 2nd one fails, same time(1st and 2nd both fail) 3rd one fails ...

assume whether a novice beat the expert or fail is not influenced by other novice or novices at all. and the chance is p

1st 2nd 3rd 4th 5th 6th 7th 8th 9th novice
  p     p    p    p    p    p     p    p    p =0.48   chance of everyone fails

Life is so boring, i agree.

i like the song Anything but Ordinary by Avril.