Analysis and Calculus

I know the Jacobian of the coordinate transformation measures how much the transformation is expanding or contracting the area around a point in G as G is transformed into R. My question is why we need the Jacobian in the transformation,the deltaA which represent small area go into almost zero in the integral,and the deltaS which represnt the transformed small area also go into zero,why we need the Jacobian to measure the multiple.

There's something about this thickness of a material to find the volume that I can't quite figure out. Let's say I have a shape with an Outer surface that's generated by rotating y=x^2 around the x axis, and its 2 inches thick all the way around, meaning that at any given point on the curve y=x^2, you can draw a line perpendicular to that point to reach a secondary curve that is only 2 inches away on the inside that has nearly the exact same shape. How do I describe this volume in terms of using a summation of hollowed cylinders? The greatest mystery of all time, only the best can solve it...cause no one can answer it...

Hi, This was posted online kind of like a challenge. I've been through calculus courses, but still, this looks weird. For example, the R doesn't seem to belong, and I have no idea what it stands for. Help?

In other words, what axiom implies 1 < 2?

I am really confused over the general idea of the summation of infinitesimals of some quantity. For example, can anyone show me mathematically how an infinite number of dm can either equal 5 kg's or 10 kg's? I can understand basic calculus.

I would do this myself, but the knowledge in math and physics goes beyond me. I have never even thought about it, but figured someone probably has. Anyway, is there anything in physics or math that goes into the probability of Political Corruption, or just corruption in a system in general. For example, the US Congress has 535 members' what's the probability of all 535 members being corrupted. Kind of like Compartmentalization I figure also, there would have to be breakage in a system if it got to big. For 535 people I have no idea. I figure it would be 1/<Insert really long number here> chance that 98% of people on the planet are in on some sort of cons…

So there's all sorts of equations where when you solve for their inverses, you don't get a "function", but I honestly couldn't care less if it's not a function, I just want to know the straight up relationship between y and x of the inverse and I don't care if it fails the vertical line test, so be it. I know there's some way to convert cartesian equations to polar equations, so can I convert, say, y=^6-8x^5+4(x+2)x^4-(x-4)x^3-20, convert it to a polar equation, solve for the inverse in terms of polar coordinates, and then back substitute into cartesian coordinates to get an x-y relationship with y in terms of x? I really don't mind if I have a bunch of roots, technically…

What's the difference between these three? I read the Wikipedia article and all I understand is:- Formal power series are infinite and do not necessarily converge Power series converge Asymptotic series is converge after finite steps Am I right? Also if we want to implement these (I mean using a computer), there are different algorithms for each? Why is that? All feel like they do the same thing but I read many algorithms to compute formal power series, asmyptotic expansion etc. Why?

No, it's not a homework problem, it's a random thing I was working on when playing around with algorithms to convert between music and systems of math. I have these terms, [MATH](1/1)x+(1/3)x^3+(1/15)x^5+(1/105)x^7+(1/945)x^9[/MATH]... And I'm trying to come up with a formula to describe the coefficients in terms of a summation as part of a larger formula that I've already broken down. I've spent more than two hours testing different formulas and none of them work, but I know the concise pattern. If you want the nth coefficient, you'd do 1 divided by the derivative of the nth term times the coefficient (n-1)th term times the coefficient of the (n-2)nd term times…

I am not in multivariable calculus yet, but I am going crazy trying to create the formula for a cylinder by integration, not a homework assignment. I can't put h in terms of r; is there another way to do this using single variable calculus? If the answer is no, then is the inability to put one variable in terms of the other mean that I must use multivariable calculus?

I am learing directional derivative and one context in the book puzzle me, i know delta f should be divided by delta s not delta x when we use directional derivative,but i really can't understand the explain in the book. Anyone can help me ?

2+2=4 =4-9/2+9/2 =square root of(4-9/2)2+9/2 =square root of(16-36+81/4)+9/2 =square root of(-20+81/4)+9/2 =square root of(25-45+81/4)+9/2 =square root of(52-2*5*9/2+(9/2)2)-9/2 =square root of(5-9/2)2-9/2 =5-9/2+9/2 =5 Hence proved that 2+2=4 i have mistaken anywhere then please tell me!!!!

Is it mathematically correct to say √16=±4, or would it be correct to say that √16=4, unless specified ±√16? This isn't for any specific problem or anything, but I'm just curious if is correct to have ±, because I know every number has two square roots.

A variable line through some point P(j, k) intersects the x and y axis at points X(a, 0) and Y(0, b). You are to find the minimum are of the triangle XOY. After a few pages of scratch paper, I gave up and looked up the solution. I got as far as determining that the slope of the variable line would determine the area of the triangle so I used the 'point-slope' equation for the line. What I don't understand is why is equating x and y to zero in the equation of the line valid? The way I thought about it is when x (or y) is zero, then the line would sort of shoot off to the right and create a line that goes through the Origin and the point P(j, k) which would b…

If on a test or experiment you result above the average of the sample group that is generally regarded as good. However, what if you score above average, but the average is failry low...say out of a sample size of 1000, the average score is 40%, and your "above average" score is 49%, with the highest score being 59 (standard deviation is 10.7). Is this "above average" score really that great or is it just a little bit better? (these are not actual sample scores, just random numbers)

Suppose we have a function consisting of a series of matrices multiplied by a vector: f(X) = A * B * b --where X is a vector containing elements that are contained within A, b, and/or b, --A is a matrix, B is a matrix, and b is a vector Each Matrix and the vector is expressed as more terms, ie... X = (x1, x2, x3) A = [ x1 + y1 y4 y7 ] [ y2 x2 + y5 y8 ] ] y3 y6 x3 + y9 ] B = [ y1 x2 + y4 x3 + y7 ] [x1 + y2 y5 y8 ] ] y3 y6 y9 ] b = [y1 y2 y3]' (' means transposed) Now we want to find the Jacobian of f - ie the partial derivative of f wrt X. One way to do this is to multiply the two matrice…

You are given F(v) = -Cv2, the force as a function of velocity, and you need to show that x - x0 = (m/C) ln(v0/v) where m = mass and C is some constant. My question is why and how does definite integration work when you have 2 variables? I mean I was able to finish the problem using this as I've seen it be used in the textbook itself and also some math books, but I don't really understand or know how it works. In calculus classes, and also my book, definite integration is done using 1 variable. I mean we do multiple definite integrations but that's only 'on one side' and 'one at a time' and I've never really been taught definite integration on differ…

Calculation of [math]\displaystyle \int\frac{1}{\left(1+x^4\right)^{\frac{1}{4}}}dx[/math] My Trial : [math]\displaystyle \int\frac{1}{x\cdot \left(x^{-4}+1\right)^{\frac{1}{4}}}dx[/math] Let [math]x^{-4}+1 = t^4[/math] and [math]\displaystyle \frac{1}{x^5}dx = -\frac{4t^3}{4}dt = -t^3dt[/math] [math]\displaystyle [/math] Now How can i solve after that Help please Thanks

how do we find the square root of a recurring decimal eg) 1.3333*. to what level of accuracy can we predict the square root, cube root, nth root etc?

i'm not sure if i'm right, i know there's another method in solving this but i'm not sure how to do that

An isosceles triangle is inscribed in a circle of radius R. Find the value of theta that will mazimize the area of the triangle

Is [math]z^{\frac{1}{2}}[/math] equivalent to the principal square root or the plus-minus square root?

hi ! I have some trouble calculating this product: S=(1-x^k) * (1-(x+1)^k) * (1-(x+2))^k * (1-(x+3))^k .... (1-(x+n))^k I know that ln(1-x) = x (approximatively) When applying the logarithm I get ln(S)= x^k + (x+1)^k + (x+2)^k + ... + (x+n)^k So: S= exp (x^k + (x+1)^k + (x+2)^k + ... + (x+n)^k) I want to bound this but I don't know how to do it from here. Thank you !

Hello, I'm new to this forum so if I'm breaking the rules by posting in the wrong place etc then please correct me. I'm a British student studying Chemistry in Chinese in Xiamen University in China. I've been re-capping my Chinese language for the past year and nI have just started my degree this year. I only have two subjects this semester - Calculus and Inorganic Chemistry, whilst the Chinese students have additional subjects: Marxism, Army training, etc. My point is that I havn't studied Mathematics or Chemistry for a while so I need some help, hence why I joined this forum. Here is my problem: Prove that if c>0 Then \lim_{x-c}\sqrt{x}=\sqrt{c} …

I want help correlating two sets of data. Let's assume a publisher prints 100000 copies of a book. The size of that book is 1131 square mm and the cost of placing an advert on the back cover of that book is $3415.15. If another publisher decides to print 5000 copies of the same book, number of pages being same but the size reduces to 211.05 square mm, what would be the advert rate on the back cover? i,ll appreciate the mathematical basis for any solution. Regards