Scalar, vector and tensor calculus

In summary, there is a parallel between scalar and vector calculus, as seen in the equations ##v=at+v_0## and ##\vec{v}=\vec{a}t+\vec{v}_0##, and the equation ##s=\int v dt = \frac{1}{2}at^2 + v_0 t + s_0## can also be written in terms of tensor calculus. The definitions in calculus for vectors and tensors are based on those for scalars, so it is natural for there to be similarities in the equations used. There are also formulas that have scalar, vector, and tensor versions, such as ##F=ma##, ##\vec{F}=m\vec{
  • #1
Jhenrique
685
4
I noticed that sometimes exist a parallel between scalar and vector calculus, for example:

##v=at+v_0##

##s=\int v dt = \frac{1}{2}at^2 + v_0 t + s_0##

in terms of vector calculus

##\vec{v}=\vec{a}t+\vec{v}_0##

##\vec{s}=\int \vec{v} dt = \frac{1}{2}\vec{a}t^2 + \vec{v}_0 t + \vec{s}_0##

So, this same equation could be written in terms of tensor calculus? Or exist some equation that can assume a scalar, vector and tensor form?
 
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  • #2
I'm not sure what you intend here. The definitions in Calculus as extended to vectors and tensors are done in imitation of scalar Calculus so of course you have the same formulas.

(I would not consider your equation "of vector calculus" to actually be "vector Calculus". Your coefficients are vectors but your variables are not.)
 
  • #3
HallsofIvy said:
The definitions in Calculus as extended to vectors and tensors are done in imitation of scalar Calculus so of course you have the same formulas.

You can give me an example of some formule that have a scalar(rank0), a vector(rank1) and a tensor(rank2) version?
 

Related to Scalar, vector and tensor calculus

1. What is the difference between a scalar, a vector, and a tensor?

A scalar is a quantity that has magnitude but no direction. A vector has both magnitude and direction. A tensor is a mathematical object that can represent a linear mapping between vectors, or a multi-dimensional array of numbers that follows certain transformation rules. In simpler terms, a scalar is a single number, a vector is a combination of numbers with a specific direction, and a tensor is a more complex mathematical object that can represent multiple vectors.

2. How is scalar, vector, and tensor calculus used in science?

Scalar, vector, and tensor calculus are used in various fields of science, such as physics, engineering, and mathematics. They are used to describe and analyze physical quantities, such as velocity, acceleration, and force, and to solve complex equations and problems involving multiple variables and dimensions. They are also essential in understanding and studying the behavior of physical systems, such as fluid dynamics and electromagnetism.

3. What are some common operations in scalar, vector, and tensor calculus?

Some common operations in scalar, vector, and tensor calculus include differentiation, integration, and the dot and cross product. Differentiation is the process of finding the rate of change of a function, integration is the process of finding the area under a curve, and the dot and cross products are used to calculate the magnitude and direction of vectors, respectively.

4. Can you give an example of how scalar, vector, and tensor calculus are used in real life?

One example is in the field of fluid dynamics, where vector calculus is used to describe the flow of fluids and the forces acting on them. Scalar calculus is used to analyze the pressure and temperature of the fluid, while tensor calculus is used to describe the stress and strain of the fluid as it flows through different dimensions and directions.

5. What are some resources for learning scalar, vector, and tensor calculus?

Some resources for learning scalar, vector, and tensor calculus include textbooks, online courses, and video tutorials. Some recommended textbooks are "Vector Calculus" by Jerrold E. Marsden and Anthony J. Tromba, and "Tensor Calculus" by J.L. Synge and A. Schild. Online courses are available on platforms such as Coursera and edX, and video tutorials can be found on websites like Khan Academy and YouTube.

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