Intro
Numerical integration, algebraic operations, integration, differentiation, derivatives, matrix and determinant calculations, these mathematical knowledge can be understood by knowing the basic principles of calculation. I don't know why universities always give us difficult questions to deliberately make it difficult for us. In practical cases, integration or differentiation problems are often solved using computers.
So how can we use computers to help us? Computer programs like MATLAB, Mathematica, plugins for office, and the Edge web browser can all help us quickly solve mathematical problems, but these tools are limited to computers. Today, I will recommend a very powerful online tool - Wolfram Alpha.
Open the WolframAlpha official website
We can see that the page is relatively clean, but there are many listed functions, including mathematics, science, social culture, and daily life. Each section contains many subdivisions. Personally, I think the most practical one is mathematics. If I had known about this tool when I was studying calculus in my first year, I wouldn't have scored so low.
Mathematics
Now I will briefly show you its functions.
Solving ordinary definite integrals, indefinite integrals, determinants, and matrix calculations are the most common functions of calculation tools. I will only demonstrate the following two functions.
Solving differential equations has always been the most troublesome for me. Besides the results, this program also plots relevant graphs. When the step-by-step option is selected, the program will show the calculation process step by step to help everyone understand how it is calculated.
Another function is plotting function graphs, especially when calculating multiple integrals and higher-order equations, the functions can become very abstract and difficult to understand. The plotting function of this program can help us understand to some extent. Here is a simple example.
Science and Engineering
I only demonstrated chemistry and fluid mechanics. In terms of chemistry, you can search for properties of elements, atoms, ions, etc. For example, "10 densest elements" can also search for properties of compounds and convert them to the amount of substance based on mass or volume. Here is a demonstration of a molecular formula.
For fluid mechanics, I only provided an example of streamlines when a flow field passes through an object.
There is also a very useful function, which is data fitting. The operation is also very simple. By inputting existing data, it can automatically plot the fitted graph and manually modify the function type, whether it is a linear function or other function types.
Encyclopedia
In addition to calculations, you can also use it as an encyclopedia to directly search for the content you want. Here is an example of searching for celebrities.
ChatGPT
Finally, the most important point is that Wolfram Alpha is one of the collaborators of the ChatGPT official plugin. Users of gpt-4 can directly use this program in ChatGPT to generate function graphs and solve differential equations. If you are interested, you can subscribe to gpt-plus to experience this plugin. I believe it will definitely give you a different experience.
Summarizing
There are more and more diversified and convenient calculation tools, and the threshold is getting lower and lower. This makes me wonder if our education model should make changes. When will we stop giving students difficult problems to make it difficult for them? Similarly, the birth of AI and GC tools is also to allow ordinary people to use these tools with lower thresholds. It is hard to imagine that in the future, people who do not use AI and GC tools may be like the older generation who did not use smartphones a few years ago.