Transcription
If you study engineering or physics, you've probably come across tensors. But what exactly are tensors? And why do tensors often look like mattresses? Even though mattresses and tensors are not quite the same. That's exactly what we will discuss in this video.
First, I'll give you two simple reasons why tenzos are not just matrices. And at the end, we will look at an example that shows why we need tenzos in the first place.
First of all, you can think of tensors as the generalization of the concepts of scalas, vectors and matrices. A scala is just a number. We also call this a zeroorder tenzor or rank zero tensor. A vector is an array that contains multiple numbers in a column which we also call first order tensor. And a matrix which is an array of numbers with rows and columns is called a second order tensor. Now we can extend this idea to third order tenzos by adding another layer of numbers to the array. As we add more and more layers, this becomes more and more difficult to visualize. So one way to get around this is by using index notation. The components of the vector may be denoted by vi where i is either one or two in this two-dimensional example. The components of the matrix can be denoted by fi j where i and j are either one or two. and the components of the third order tensor can be denoted by t i j k and so on. Index notation is useful for several reasons. For example, it's quite convenient to manipulate equations in index notation. And it's also nice that we can immediately see the order of the tensor. Zero order tensors have zero indices. First order tensors have one index. Second order tenzoros two and so on. So to summarize, one reason why matrices and tenzos are not exactly the same is that matrices always have two indices, but tenzoros can have arbitrarily many.
But this doesn't tell the whole story. There's a second probably more important reason why tenzoros are more than just higher order matrices. And this is best explained with an example. Let's consider this vector which as I said can be interpreted as a first order tensor. The components of the vector tell us how the vector looks like in a Cartasian coordinate system. But here's the point. If we change the coordinate system, the vector itself does not change. But the components that describe the vector in the coordinate system change. The area of numbers on the left is just one representation of the vector in a particular coordinate system. If I just give you these numbers, you don't know how the vector looks like unless I also tell you the corresponding coordinate system. Tensors are mathematical objects that are independent of the coordinate system. The first order tensor that we see on the right doesn't change if we change the coordinate system. But the numbers describing the first order tensor change. So you could say that this is not the tensor. This is the tensor and this is just one way to describe the tensor in a particular coordinate system.
Okay, let's apply what we have learned about first order tenzoros to second order tenzoros. Let's take this matrix as an example. A matrix is difficult to visualize. But what we can visualize is what happens when we multiply the matrix by some vectors. For example, multiplying the matrix by this vector gives us this transformed vector. And we can illustrate this transformation for any arbitrary vector. Another way to visualize this is to draw a circle around the arrow tips which stretches as the arrows are transformed by the matrix. You can think of the stretched circle as a graphical representation of the second order tensor. As a side note, the word tenzor comes from the Latin word tender which means to stretch. Now what happens if we change the coordinate system? the stretch of the circle does not change which means that our second order tenzo doesn't change but the area of numbers describing the deformation in the particular coordinate system changes and this is exactly why we say that matrices and tensors are not exactly the same. The tensor is an abstract mathematical object that describes the stretch of the circle independent of the coordinate system and there are infinitely many matrices that can describe this deformation in specific coordinate systems. So the next time you see a tensor represented by a matrix, try not to think of the matrix itself as the tensor. Instead, recall that this matrix is just one way to express the tensor in a particular coordinate system. The tensor itself is a more abstract mathematical object that describes the deformation of the circle independent of the coordinate system. Of course, in many engineering applications, we work with a fixed cartasian coordinate system. In this case, there's just one unique matrix that describes the tensor. And usually, we don't have to worry too much about the difference between matrices and tensors.
Finally, I want to show you an example that explains why we actually need tensors. Imagine you want to analyze the deformation of an object. It could, for example, be a solid material or a fluid like this drop of water. Tensors are useful for describing such deformations. Let's take a look at this in 2D. One way to describe the deformation mathematically is to draw a set of points on the object and track their motion over time. The so-called displacement vectors indicate how each point of the object shifts relative to its original position. But the displacement vectors cannot really tell us how the object is deforming at each point. Take this deformation as an example. Some of the points are not moving at all. At these points, the displacement is zero, although the object is clearly deforming. This means that the displacement doesn't provide information about the local deformation of the object. So, what we can do instead is to draw a bunch of small circles on the object and track that deformation over time. In this way, we can see the local deformation at each point, even if the points themselves are not moving. You might already notice that the way the circles deform looks a little bit like the deformation we saw when visualizing second order tensors. In fact, at each point, the deformation of the small circles can be described by a second order tensor. And it's important that the deformation of the small circles does not change if we change the coordinate system. The deformation and the second order tensor describing the deformation are independent of the coordinate system. only the matrices describing the tenzoros change. Again, if we work with a fixed coordinate system, we don't have to worry too much about this.
That's all I wanted to show today. I hope you have developed some intuition about tensors and why we need them. Let me know if you have any questions. Don't forget to like and see you soon. Bye.