Aus der Schule kennst du doch bestimmt noch Aufgaben/gleichungen wie : 4x=8. Die musste man lösen und man hat dann durch 4 geteilt und hat x=2 bekommen. Das war 1 Gleichung mit 1 variable.

Dann kamen Aufgaben mit 2 Gleichungen und 2 Variablen

Zum bsp 4x+2y=8 und 3x+2y=6. Jetzt ist es Aufgabe ein wertepaar (x/y) zu finden sodass beide Gleichungen erfüllt sind x=2, y=0, Wichtig hierbei ist,dass die Lösung für beide Gleichungen funktioniert. Zum bsp funktioniert das wertepaar x=0, y=4 nur für die erste genannte Gleichung. Jede Gleichungen für sich allein betrachtet hat unendlich viele Lösungen. Nur wenn man sagt es sollen beide gelten gibt es hier(gibt auch andere Fälle) genau eine Lösung die für beide Gleichungen gleichzeitig funktioniert.

Jetzt kann mam das beliebig hochschrauben. Man nimmt 3 Gleichungen und 3 Variablen. Wieder sollen alle diese 3 Gleichungen für ein wertepaar(eigtl.tripel) Gelten (x/y/z) .

Zum bsp könnten x y und z Einkaufspreise sein.

1 Person kauft 3x+4y+7z= 20 und zahlt 20 euro und wenn man den Einkauf noch von zwei weiteren Leuten kennt(sie kaufen nur die 3 genannten Produkte) kann man nun ausrechnen wie teuer die Produkte waren. Das ist jetzt zwar nicht der klassische Anwendungsfall aber man kann es sich so ganz gut vorstellen

Die Antworten lesen sich recht philosophisch, daher hier mal eine etwas stumpfere Antwort:

Ein Gleichungssystem (egal, ob linear oder nicht) ist eine Ansammlung an Gleichungen mit mehreren Variablen (du kannst es als mehrdimensionale Abbildung auffassen), die du alle gleichzeitig zu lösen versuchst. Hast du ein solches System also aufgelöst, erhältst du die Menge aller Kombinationen an Werten für die einzelnen Variablen, die dafür sorgen, dass alle diese Gleichungen gleichzeitig wahr sind.

Im Fall von linearen Systemen kannst du sogar etwas konkreter werden: Du suchst zu einer gegebenen Matrix einen Vektor, sodass nach Matrixmultiplikation ein entsprechender anderer Vektor herauskommt. Wozu braucht man das? Lineare Algebra ist der Grundstein für eine riesige Menge an Bereichen, von klassischer Mechanik und Elektrodynamik über Quantenmechanik und Optik zur allgemeinen Relativitätstheorie. Überall lassen sich Gleichungssysteme aufstellen, und überall lassen sich die fundamentalen Ideen, die man beim Lösen von LGS anwendet, weiterverwenden.

Wann immer du einen Sachzusammenhang mit mehreren Bedingungen (=Gleichungen) hast, findest du durch das Lösen des Gleichungssystems Werte für die Variablen, sodass alle Bedingungen erfüllt sind.

Das hängt davon ab von welchem Blickwinkel du das betrachten möchtest.
Nach dem ersten Semester Mathestudium würde man sagen, man rechnet mit Matrizen.
Wenn man Mathematik ganz funktional als Modell der Wirklichkeit betrachtet würde man das wohl abhängig machen von den Informationen die man da gerade mit einem Gleichungssystem modelliert. Dann würdest du wenn du ein Gleichungssystem löst, z.B. die Fahrzeit deines Autos auf dem Weg von A nach B berechnen.
Wenn man Mathematik fundamental betrachtet, dann nutzt man einfach verschiedene Eigenschaften der Identiätsrelation aus um neue Identiätsausagen aufzustellen. Transitvität, Symmetrie, Reflexivität usw.
Ich als Philosoph würde einfach die Antwort geben, dass du aus einer Menge von Informationen die du als wahr akzeptierst, also Prämissen, eine Information steng logisch erschließt. Du stellst also beim "auflösen" eines Gleichungssystems nur ein deduktiv gültiges Argument auf.

Du suchst den Schnittpunkt von zwei Geraden.

Eine lineare Gleichung kann als Gerade in einem Koordinatensystem dargestellt werden.

Zwei Gleichungen sind zwei Geraden - die treffen sich irgendwo, sind parallel oder gleich.

Praktisch können das Kosten sein, oder Wegstrecken oder Kräfte - irgendwo sind sie gleich und dieser Punkt ist oft interessant, weil sich dort die Praxis ändert: erst war das eine teurer, danach ist es das andere.

This clip explains it https://youtu.be/S31xk9PFSp8?si=Um4pTbiYdPz6CYpX&t=13

OK, I'll give you a short answer but first I'll go through a bit of physics.

For the simplified but usable model we assume that the forces that are working against the turning boat are both the lateral water resistance and also the boat's inertial momentum or the mass's resistance to rotation.

Below is a derived formula that takes into account the boat's geometric shape and mass, and also the point of attachment's location. I'll spare you of the actual calculations and give short answers in the end.

We assume that the body is a half-submerged sideways cylinder with lateral resistance of cw=0.5

{[L*d*rho*cw*(l*alpha*t)^2*0.5] + [0.25*M(0.5W)^2+0.33*M*L^2]*alpha}*(2/W) = force, in rope, in Newtons

Where L is the length of the boat, d the submerged depth (half of width W), rho is water density (1000kg/m3), cw lateral resistance of a cylinder = 0.5, l the half the length of the boat (for average lateral speed), alpha the angular acceleration ( = 2*turned_angle/time^2 = 1/6 rad/s^2 for our case) and M the mass of the boat (kg).

I used L=5 m, W=2 M and M=1000kg for the motorboat in the video and got 1356 N for the water resistance and 1430 for the angular resistance, and 2786 N for the overall engine thrust and/or the pull of the rope. That's about 280 kg or 617 lbs of maximum momentary pull of the rope which seems reasonable because the rope is cleated and is seen very taut for a short period.

They might have used a 1/2" nylon rope with tensile strength of 5750 or 2300 kg which should be OK.

For a large sailboat I used L=12 m (40'), W=3 m and M=10000 kg. With a large sailboat we will want to turn much slower (let's say 30 seconds to video's 5 seconds). This increases the water resistance only a little (as it's dependent on the square of the velocity), to about 1.9 kN. But as the boat is a lot bigger, the inertial resistance goes up to 13.4 kN. This gives about 20 kN or 2000 kg or 4400 lbs for the overall thrust/pull of the rope.

If you use a 3/4" nylon rope with tensile strength of 19000 lb or 8000 kg, you should be all right with strength to spare.

I guess you could derive a shorter heuristic formula from all the stuff above but I'll leave that to others :)

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2(5.25) + 3d = 19.05

8(18) + d = 219

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Everything is a harmonic oscillator if you're brave enough.

Whenever you want to study (small) perturbations out of an equilibrium, you will almost certainly end up with a quadratic equation somewhere.

My favourite use is solving 2nd order partial differential equations to yield a complex number that provides the frequency and damping factor of a damped harmonic oscillator.

So there are quiet a few quadratic functions in physics. E.g. if you have free fall (time is quadratic there) or gravity and electromagnetism. But really the reason you have it in the curriculum is, its pretty much the last polynomial you can solve easily by hand, so a lot of approx. And techniques boil down to get it at least to a quadratic equation and solve that.

This is awesome! This is exactly the kind of answer that kids need to that question. Keep up the good work!

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Law of cooling?

Took me one google search

It depends on exactly what you’re looking for. You can take any applicable mechanical problems that we see in life like pendulums and circuits and show how those are integral to design since circuits are in literally everything nowadays. You could also take things like the design of systems in general, as we use control systems (think your ABS module in your car, traction control, etc, any system with a control in it) to regulate and monitor everything as well. Sorry I don’t have anything extremely specific, but Diff eq and calculus IRL applications are something I’ve only scratched the surface of and am very interested in learning more about.

A bunch in economics. Ex, Individual consumer choice functions take commodity vectors as arguments. A lot of optimization problems in economics that involve matrix valued functions have you evaluating Hessian matrices (maximum likelihood estimation of econometric models for example).

Computer graphics and video games. Your computer is constantly performing linear algebra to render things to your screen. It's super cool!

Page rank using eigenvalues , I believe that’s how certain search engines such as Google started out. Of course the Algorithm has gotten complex and more efficient, but if I remember correctly, I believe they used eigenvalues/vectors to find pages that matched the search the user was looking for and ranked them thus giving more relevant results

Machine learning. Atomic orbitals and quantum physical chemistry. Figuring out voltages in every part of a complex electrical circuit. Those are the top three things I have used LA for in grad school.

One of the first example problems in my first year linear algebra textbook had to do with making trail mixes of various values by weight given the cost by weight of the ingredients.

It's also used for statistics, which itself is applied to a wide variety of real life scenarios.

have you tried typing it into google and clicking the first result?

https://www.google.com/search?q=applications+of+integration

->

https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Calculus_(OpenStax)/06%3A_Applications_of_Integration/6.5%3A_Physical_Applications_of_Integration
https://www.math24.net/integrals-electric-circuits
https://www.math24.net/applications-integrals-economics
https://www.intmath.com/applications-integration/applications-integrals-intro.php
https://tutorial.math.lamar.edu/classes/calci/intappsintro.aspx
https://mathstat.slu.edu/~may/ExcelCalculus/sec-7-8-BusinessApplicationsIntegral.html
https://www.quora.com/What-are-some-real-life-applications-of-integration-and-differentiation
https://homepage.divms.uiowa.edu/~stroyan/CTLC3rdEd/3rdCTLCText/Chapters/Ch14.pdf
https://math.stackexchange.com/questions/587840/is-there-a-practical-real-use-of-integration

Well integrals don’t mean the rate of change first of all. The easiest way in my opinion to conceptualize an integral is by thinking of it as an antiderivative–I want to know what some function is the rate of change of, so I integrate–or you can think of it as the area under a curve. As I’ve gone through more and more math and physics, I’ve also realized that it’s essentially a method to add up a bunch of values that are all different.

A classic example is velocity and displacement. If my velocity is described by the curve x^2 how far have I travelled in, say, 20 seconds? What makes this question tricky is the fact that our velocity isn’t constant. If it were constant, I could just multiply my velocity by the amount of time I traveled and figure out the distance I went. An integral does that, but with each and every little velocity you travel at. So, for instance, an integral says “you traveled 20 mph for a reeeaaally short period of time (infinitely small in fact because velocity is always changing), then you traveled 20.00001 mph for a really short period of time, then 20.00002 mph for a really short period of time...” and it takes all those little increments of velocity and all those little increments of time and multiplies them together to get a little increments of distance (because, remember, velocity*time=distance) and adds all those little distances up to get the precise distance you travelled.

Another way of looking at this is that integrals and derivatives help us get more information out of the information I’m initially given. If you give me only acceleration, I can tell you how fast the object was moving and how far it went. If you give me the force on an object, I can tell you how much momentum it has and how much energy it has.

Rate of change is actually a derivative. Equations with derivatives are solved by integration.

Integrals are sums. When you want to find the total of something discrete, you use addition. When you want to find the total of something continuous, you integrate.

For example, you may know that the equation for work done (ie, energy expended) to counteract a force F over a distance d is W=F*d. Except this equation only makes sense when F is constant, and usually it isn't. If you have a force of 100 newtons for the first meter, and a force of 200 newtons for the second meter, then the work will be W=(100)(1)+(200)(1).

In general, the force can vary continuously. So to find the total work done, the actual formula in the general case is W=∫F(x)dx. You fundamentally cannot avoid integrals if you want to do anything meaningful with physics.

If you think addition is useful, and you think continuous things exist, then integrals are useful for the same reason.

Integrals have many real-life applications, like calculating areas under curves (useful in physics for distance and velocity), determining the volume of irregular shapes, and finding the total accumulated quantities (like income over time in economics). In engineering, they're used to analyze forces and motion. Understanding these applications of integrals shows how they connect math to practical scenarios!

The x,y plane is a Cartesian product. It is R x R. It has real life applications in geometry and physics and many other places.

It’s very useful in combinatorial problems, which are relatively abundant in real life if you look for them. For example, given a set of m pairs of pants and n shirts, you can form m x n distinct outfits using these articles of clothing. Try to work out for yourself why this is true.

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Cartesian products map one set to another. Students to classes, people to homes, even a slot machine pull maps one value (a pull) to multiple outcomes of the slot machine.

Quantum mechanics. Schrodinger's equation is a partial differential equation with complex coefficients. Its solutions are complex-valued functions of time and space. The magnitude (aka modulus, aka "distance from the origin") of that solution gives you the probability distribution for the location of a quantum particle over time.

An imaginary number is a number times i (or j, j used in electrical engineering as i means current) which is equal to the square root of negative 1. A sinesodal wave can also be thought of as a 3 dimensional corkscrew/ spring like shape. So i is used as the 3rd dimension. So there is real, time, and imaginary. When you look at the real and imaginary section you would see a circle (think looking through the hole of a spring) around the center of 0 real and 0 imaginary. A point on the circle can be described via the distance from real=0 and i=0, which would form a right angle triangle with those being the sides. The same point could also be described using the hypotenuse and an angle. It is useful in electrical engineering because ac eletricry is a sinesodal wave as the voltage potential changes. Complex numbers just mean a real number plus an imaginary.

Complex numbers are used massively in electrical engineering as it is much easier to use complex numbers than sines and cosines

Complex numbers have a lot of interesting usage in real life but I’ll try to generalize as much as I can: complex numbers let us use algebraic tools to apply geometrical actions, and to write in a much more dense way.

What I mean by dense way is very common in physics where we can use complex number to write a lot of information in simple ways. We can for example use one complex expression to describe an objects location and velocity, instead of needs a longer real (I.e. not complex) way of writing both.

The second cool thing about complex numbers is that they allow us to blur the lines between algebra and geometry. Let’s say you have a triangle in the xy plane and you want to rotate it by 60 degrees. It takes a lot of hard work, and some nasty math to calculate the new position of the vertices. However, if you use the complex numbers, you merely need yo multiply each point by a single number, e^(pi/3) and you are done.

As such, complex numbers are used a lot in both physics, but also programming, design, and similar topics.

It's used extensively in differential equations. And is used heavily in mechanical and electrical engineering.
You know how when you push on a spring it wiggles back and forth? That's governed by imaginary numbers. The term imaginary number is really bad. They're very real and have an effect on the world. They're "perpendicular" to real numbers.