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uiulic
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Any interested in this topic?
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uiulic said:Question1: Why is force a vector?
uiulic said:Question2: In continuum mechanics (Malvern 1969), a second order tensor (stress tensor) is defined as a linear vector function with both the argument and output of the function are vectors. In my mind, tensor is a quantity. Why is the function is an quantity?
uiulic said:Question3: In terms of linear vector function.
uiulic said:Let’s assume is a linear vector function b=f(a), a and b are vectors. Then f is said to be a second order tensor.
uiulic said:Why don’t you say b (b has been defined as a vector already) is a second order tensor? Since we can understand b=f(a),the linear function relation, as b(a)=f(a) , and b is then a second order tensor. Where does the problem occur for this explanation?
Greetings uiulic and welcome to the forum. I'm currently in the process of creating a branch on my website to describe geometrical methods of physics. There is a page there on an intro to tensors. It gives the basics. Its located atuiulic said:Any interested in this topic?
uiulic said:Question1: Why is force a vector?
Hi Robrobphy said:(The magnitude of a vector is based on a choice of metric, which is an additional structure... and which is not part of the standard textbook definition of a vector.)
pmb_phy said:Hi Rob
I'm curious about this magnitude thing. While I do not disagree with you on this part, I do have a Dover text called Differential Geometry, by Erwin Kreyszig, in which the author defines a vector according to the simple definition I gave above. I gave the correct definition in parentheses for those more informed on the matter. In your opinion, what would be your guess as to why the author would define a vector in this way with no mention of a metric??
I this instance I chose that definition so as not to bring in higher order of mathematics that the OP may not understand.
Thanks
Pete
A tensor is a mathematical object that describes the relationship between different coordinate systems. It is a generalization of vectors and matrices and can represent complex physical quantities such as stress, strain, and flow.
The components of a tensor depend on the dimensionality of the space it is being defined in. In three-dimensional space, a tensor can have 9 components if it is a second-order tensor (a matrix) or 27 components if it is a third-order tensor.
A vector has both magnitude and direction and can be represented by a one-dimensional array of numbers. A tensor, on the other hand, can have multiple components and can represent more complex relationships between quantities. Vectors are a special case of tensors.
Tensors have many applications in physics, engineering, and computer science. They are used to describe the movement of fluids, the deformation of materials, and the behavior of electric and magnetic fields. In machine learning, tensors are used to represent and manipulate data in multi-dimensional arrays.
Tensors play a crucial role in Einstein's theory of general relativity, which describes the effects of gravity on the fabric of space and time. Tensors are used to represent the curvature of spacetime, which is determined by the distribution of matter and energy. This allows us to understand the behavior of objects in the presence of strong gravitational fields, such as black holes.