Is there any work on this kind of mod math? It's basically the usual modular mathematics, but it oscillates between positive and negative.

For instance, if we look at a clock, it would go 0 -> 11 in the positives, then 0 -> -11 in the negatives, and then back. If it's positive, it refers to am, and if it's negative, it refers to pm. E.g. 6 would refer to 6am, and -5 would refer to 5pm.

My first thought was for mod m, use m that is negative. But if I take a number A and negative m, then A mod m is just equal to (A mod |m|) - m, which is not exactly what I'm looking for.

I know I can "cheat" by using modular mod regularly and check if it's closer to 0 or m, but it doesn't

It loses some properties of modular math (for example -1 != m - 1) but looking to see what kind of results this work would result in (if anything interesting)

Bonus: the above is using two "phases", one for the positive, and one for negative. Is there any work on having more than 2 phases. We have 1 = e^2pii, and -1 = e^2pii/2. But for instance is there a possibility we could use n phases with e^2pii , e^2pii/n , e^2pii2/n ,..., e^2pii(n-1)/n

https://preview.redd.it/uwk67nmqeqed1.png?width=3872&format=png&auto=webp&s=334d7c5d85f69474f06ebf2bf2ffa6d7f056704c

as said its a conserving field so the path is meaningless but I don't know how to deal with the parameterization, if I just input it as is I get one hell of an integral that won't even fit on the screen so there must be a better way

https://preview.redd.it/iz0hc48ubqed1.png?width=745&format=png&auto=webp&s=ed57c8f176a17c5cf45f7bef96fcae656991c2ce

I'm looking for help on (b) and (c). I answered (a) by just plugging the vector components into the cartesian equation and checking that it's satisfied.

I think I solved (b), but it was a struggle. I found two points on the line g(s0, t), found a vector equation for the line passing through them in terms of a parameter k. Then I equated an arbitrary point on g(s0, t) with my vector line equation and showed that the equation of t in terms of k was consistent for all 3 components.

As far as I'm aware that works, but it was ugly, unintuitive and took about a page of algebra for the two parts. I feel like I must have missed something obvious. This is early in a vector calculus textbook and I don't think it should be using calculus methods yet. I couldn't figure out anything using equivalence scaled differences either.

I haven't gotten anywhere with (c), though I'm guessing the image is the hyperboloid exactly. Figured it might somewhat depend on (b) and I want a more intuitive answer to that first.

I did calculus maybe 15 years ago. I think I have a reasonably good recollection of what I learned, but I would like to review the subject. But I also don't want to spend to much time doing so. Do you know of any good book to just refresh your memory? Preferable with a limited number of exercises?

is there some general way of determining wether the limit of A^(n) converges as n approaches infinity, for some matrix A.

obviously it converges for the zero matrix or the identity matrix but i haven't gotten much longer.

i'm guessing that if you can show that it shrinks the space in all directions, it would converge to the zero matrix? though i'm not confident about that.

I'm being asked to use mixed numbers in context of a salary amount. I need to figure out what "six 1/9ths" of $135,000 is.

The question is "you will receive six 1/9ths of your $135,000 salary as a signing bonus. How much is the bonus?"

Can someone show how they would solve?

e^ix = cos(x) + isin(x) is well documented and understood. I was wondering (and hoping) if there was a way of expressing e^i(x^2) in a similar way? I don’t particularly fancy trying an expansion, but if somebody knows of a way of expressing this in trig terms, I’d be very grateful

I'm supposed to find the min and max of f(x,y)=x+y over the domain

https://preview.redd.it/dlf6y813joed1.png?width=2494&format=png&auto=webp&s=c794723b12588779d4fdb1b04b56943f1f2e7af4

https://preview.redd.it/74l8zkdrjoed1.png?width=1269&format=png&auto=webp&s=3e6090ff2727a36412432673764972a790cccf13

from that, I understand I'm in the first quadrant and that (I chose to focus on y) 16/x <= y <= (33-x)/2, but then when trying to build and solve a lagrangian it got complicated, I've created the following questions (I didn't put the less than or equal and changed it to just equal since we're checking boundaries) this led me a ling path which at the end of it I got a 2 kappa's that corresponds to two different pairs of points (1,16),(32,1/2)

just a huge headache, I second guess myself at every move and not confident at all, so if anyone can both help me in this specific problem as well as link to good videos/examples to follow and learn from, thanks

Here is the problem:
"Use a graph or level curves or both to find the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely. (Enter your answers as comma-separated lists. If an answer does not exist, enter DNE.)"

f(x, y) = 3(x − y)e^(−)^(x^)^(2 −) ^(y^)^(2)

I used Geogebra to graph this because Desmos doesn't like the double superscript.
There are clearly no saddle points and the max/min are roughly z = +/- 1.8

When trying to "use calculus" as directed, I'm hitting a wall with the partial derivatives.
What I have so far doesn't seem correct (2nd picture, please forgive my writing), so I'm not sure that I can continue. When subbing in F_x(0,0) and F_y(0,0), I get +/- 3, which doesn't match the graph's extrema.
I'm pretty sure the next step is to sub F_y into F_x (or vice versa?), but truthfully I'm a bit lost and I think I'm overthinking this.

Graph

My work

Hi guys, I am really new here and I am finance student looking to work for Quants. As per my goals, I want to learn advanced probability, linear algebra and calculus. What are your suggestions as to what courses, YouTube videos or any learning resources I should use to maximise my learning.

In arithmetic, x^(-a) = 1/x^(a) , but in trig (sin(x))^(-a) = arcsin(x)^(a) , and not 1/sin(x)^(a)

Same goes for x(a+b) = xa+xb, but in trig, cos(a+b)= cos(a)cos(b)-sin(a)sin(b)

Within trig, I get that the triangle ratios follow different rules, and that the trig functions are 'special' and .. functions .. instead of direct values. The reciprocal of sinx is cosecantx, which is 1/sinx, but I just don't get why we write the ^-1 exponent instead of simply directly arcsinx. The negative exponent confuses me due to the arithmetic rule stated above.

Within calculus, when you find the differential of y with respect to x for the function y=x^2, you quickly find that dy = 2x * dx, which then results in dy/dx = 2x.

When asked to find the differential of y with respect to x for the function y=x^-2 , I expected to 'just' go and be like:

y=1/(x^2)

dy = 1/(2x * dx)

dy/dx = 1/(2x*(dx)^2)

dy/dx = (2x*(dx)^2)^-1

dy/dx = (2x^-1)*(dx)^-2

HOWEVER, I found out that, using the 'binomial theorem', the answer is dx/dy = -2x^-3.

A simple explanation of why these arithmetic rules don't apply (especially irt the calculus part) like I expected them to, would be much appreciated.

Also: I'd really like to hear of more examples like the above, so I can mentally prepare for whenever I get to those math fields.

btw: I'm a tax lawyer with just a hobby-like interest in math, so I just read books of Spivak and Kingsley Augustine in my spare time and try to do all exercises and examples they mention before I've seen the solution. Go easy on me please :)

In an art museum, there are 8 different paintings by Picasso, 5 different paintings by Van Gogh, and 3 different paintings by Rembrandt. The curator of the museum wants to hold an exhibition in a hall that can only display a maximum of 7 paintings at a time.

The curator wants to include at least two paintings from each artist in the exhibition. Given that 7 paintings will be displayed, determine how many ways they can be selected.

I tried to solve this by using 8C2 × 5C2 × 3C2 x 10C1 = 8400 ways (i.e. choose two paintings from each artist, and then choose any one painting from the remaining 10 paintings).

However, the actual solution creates 3 cases and adds all the ways of doing so together, getting an answer of 2800 ways:

• 3 Picasso, 2 Van Gogh, 2 Rembrandt
• 2 Picasso, 3 Van Gogh, 2 Rembrandt
• 2 Picasso, 2 Van Gogh, 3 Rembrandt

I understand how the actual solution works, but I also don't understand why my method doesn't work. Could someone please explain to me? Thank you!

I have solved the first part of the following problem; my question is whether the series solution converges.

My thought process is to:

• Use the Ratio test for each of the two series obtained as a solution to the ODE, and show that L<1

Is this the correct way to sufficiently prove convergence, or does this problem require a different way?

Below is my attempt to show that the first series converges.

https://preview.redd.it/sm2g3cjmtned1.png?width=1793&format=png&auto=webp&s=c542dcf676ead6a2617b8c6a84025c3da700d854

https://preview.redd.it/hf1asxantned1.png?width=1423&format=png&auto=webp&s=a7e8dd2dd95350fce689411205f1dd4ddc992cb1

https://preview.redd.it/49n4y4kntned1.png?width=1267&format=png&auto=webp&s=e7bf0be4b85c95e6e48d17c312a4789244247962

https://preview.redd.it/nh2dh9qmkned1.png?width=1545&format=png&auto=webp&s=173ab7313fd4d2354897deebc6c71e253b9ba3d4

TMUA-2023-paper-2-worked-answers.pdf (uat-wp.s3.eu-west-2.amazonaws.com)

I'm supposed to find the critical points and sort them (min, max, saddle) of the function

f(x,y)=x^2-2x+6+2y^2-4y

when the constraint is the region enclosed by the points (0,0), (8,0), (0,8) creating a right-angle triangle.

I don't know how to extract a constraint out of this

Okay. Let's say I'm spending $1,580,000 every month. However, I'm only earning $730,000 per month. It's a rate of decline of 116.4% I started with $125,000,000

The issue I'm having is figuring out how long will it take until I run out of money? And how is that calculated?

So a student came to me today and asked why I wrote the √9 as just 3 and not ±3. I gave some fluffy on the spot answer but it has now haunted me for the entire day. Who is correct here? I explained that if we start with x^2=9 then our answer is ±√9 which gives us ±3, but because we've started our equation at x= √9, that negative answer is removed.

Any assistance on this would be great!

I'm trying to make an animation of a square rolling on a flat surface, which I don't want the square to intersect. I've thought about it for a while and all I need is to move the square a distance r(θ), where r(θ) is a function describing the distance from the centre of the square to the edge going at a certain angle θ.

I've tried to solve it, but have been unsuccessful; my first attempt was 1-(abs(sin(θ)))/x and tried adjusting x, but that just leads to the square having its sides bent inwards, or blown outwards; my second attempt was trying to use Wolfram Alpha, but it gave me a function of x, y, and r, but I only want a function r(θ).

Simplifying the problem, I'm trying to find the polar equation for a unit square in terms of θ.

I've put a diagram on this post if it helps.

https://preview.redd.it/cj3v90pfomed1.png?width=546&format=png&auto=webp&s=164ff9a77b2889681eb34436463f5ab1e33f9d4c

Also, if possible, tell me if the solution also works for larger or smaller squares.

I am failing to fully understand the purpose of using codomain. I understand that the the codomain is all possible results that could come out of the function while the range is the actual results that come out of the function.

I seem to not understand the significance of using codomain and why we cannot just make it as broad is as possible. Is there incentive to find out the codomain of a function before calculating its range?

Has anyone here read “Higher Algebra” by Hall & Knight? Or anyone with information about whether the book is good for a high school student or not? Primarily using it for Olympiads and other university entrance examinations!

Furthermore, any other book suggestions are also welcome!

Thank you!

(Also, my bad if I can’t ask this here. I didn’t see anything against this in rules, so yeah. 😅)

https://preview.redd.it/7pvxu9gmyled1.png?width=549&format=png&auto=webp&s=4d94f9df10ffc2adc30a2e84b9e1663c5cdba6c9

I outlined the step I'm confused about in blue. I understand all the derivative stuff. I'm not sure how 6x^5/5y^4 can be simplified to 6y/5x, asked my professor and she basically just restated the problem. I get the concept of why you would multiply by xy/xy because it's 1 but I don't really get the steps. Appreciate any help thanks

Follow up question, why is Euclid's formula specified to only solve for even perfect numbers?

https://preview.redd.it/15yhvhce5led1.png?width=360&format=png&auto=webp&s=b57559df5c7ffc8318e4341b686f1ed2efddfb78

hi so this is part of the method to find the area of a triangle. i tried using distributive properties of cross product but i dont get the ans in the second line. any help would be greatly appreciated. thanks

For example, taking an integral is a “linear operator” despite not necessarily having anything to do with straight lines at all, and “linear algebra” deals with high dimensional vector spaces, not just unidimensional lines. I understand the definition of a “linear map,” but I’m curious about the historical or conceptual reasons for why this came to have the adjective “linear.” Arbitrary 2D lines y = mx + b on a plane aren’t even necessarily linear maps (unless the intercept is zero) so I’m curious about the etymology.