What Is the Physical Meaning of These Vector Calculus Concepts?

In summary, the conversation discusses the physical interpretations of various mathematical concepts such as curl, divergence, directional derivative, and gradient in relation to their use in physics. It is important for mathematics students to develop both rigorous understanding and physical intuition when learning these concepts. The conversation also draws a comparison between using a computer's graphics card for faster computations and utilizing our innate ability to understand and apply physics concepts.
  • #1
shermaine80
30
0
Hi,

May i know the physical meaning of the following:

(1) Curl of a vector field A(x,y,z)
(2) divergence of a vector field A(x,y,z)
(3) directional deriative of G(x,y,z)
(4) gradient of a scalar field G(x,y,z)
 
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  • #3
Tac-Tics gives possible interpretations of those in a particular application. There is NO general "physical meaning" of a mathematical concept- mathematics is not physics.
 
  • #4
These concepts were originally developed because of physics, and even the purest mathematics student should develop the physical intuition of it, while simultaneously developing the topic rigorously.

If you get a computer with a killer graphics card that can do floating point matrix computations really fast, then it would be stupid to write a program that uses the CPU instead. As humans, we have a killer "graphics card" and "physics card" built into our mind - it would be stupid not to use it.
 

FAQ: What Is the Physical Meaning of These Vector Calculus Concepts?

What is the physical meaning of gradient?

The gradient can be thought of as a measure of the rate of change of a physical quantity. It tells us how much a quantity changes over a certain distance or time.

How is gradient calculated?

The gradient is calculated by taking the derivative of a function with respect to its independent variables. In simpler terms, it is the change in the output of a function divided by the change in the input.

What is the significance of gradient in physics?

The gradient is an important concept in physics as it helps us understand the behavior of physical quantities such as velocity, force, and temperature. It can be used to describe the direction and magnitude of changes in these quantities.

How does gradient relate to slope?

Gradient and slope are closely related concepts. In one dimension, the gradient is equivalent to the slope of a line. In higher dimensions, it represents the slope of the tangent to a curve at a specific point.

What are some real-world applications of gradient?

The concept of gradient has many practical applications in fields such as engineering, physics, and economics. It is used to solve optimization problems, model fluid flow, and analyze the behavior of electromagnetic fields, to name a few examples.

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