- #1
misogynisticfeminist
- 370
- 0
I have been thinking about this for a while, but why is the del operator a vector?? The book i have states no reason why and i was thinking if you guys could tell me why.
Thanks...
Thanks...
Hurkyl said:Riddle me this: if I define [itex]\vec{f}(s) = (s, s^2, s^3)[/itex], why is [itex]\vec{f}[/itex] a vector? Doesn't the same answer apply to del?
misogynisticfeminist said:That vector f has a direction? Or because of the arrowhead? I'm really not too sure...
robphy said:Note that [tex]\vec \nabla[/tex] satisfies the Leibniz rule:
[tex]\vec\nabla(\psi\phi)=\psi\vec\nabla(\phi)+\phi\vec\nabla(\psi)[/tex]
whereas a vector [tex]\vec v[/tex] does not
[tex]\vec v(\psi\phi)=\vec v \psi\phi[/tex]
misogynisticfeminist said:hmmm, but from what I know, the definition of del is,
[tex] \nabla = \frac {\partial}{\partial {x}} \hat {x} +\frac {\partial}{\partial {y}} \hat {y}+\frac {\partial}{\partial {z}} \hat {z}[/tex]
Why the need for the unit vectors? And how does del differ from the total differential of a function?
misogynisticfeminist said:Hmmm, actually one thing which puzzles me is the gradient operator,
if the gradient operator operates only on scalars and the del is a scalar but a vector differential operator. Why is the del being a vector differential operator allowed to work on a scalar function? And why is the gradient using the del of a scalar function a vector?
Is my understanding wrong? I can't understand the terminology of tensors yet though, sorry, so i was wondering if this can be explained in a way without tensors.
misogynisticfeminist said:Why the need for the unit vectors? And how does del differ from the total differential of a function?
And note that vectors don't even necessarily have to be representable as n-tuples. Calling something a vector just means that it's an element of some vector space.
Crosson said:This statement is contradictory. All vector spaces have basis, and any element of a n-dimensional vector space can be written as an n-tuple.
mathwonk said:no offense meant here to the man well intended attempts to answer, some correct, but it is very confusing. the language is hard to get straight, for one thing,k i.e. the main thing. people are calling del something different from d, when apparebntly thet are intended to be essentially the same thing.
All vector spaces have basis, and any element of a n-dimensional vector space can be written as an n-tuple.
Del is a vector because it represents a mathematical operator that operates on a vector and produces another vector as a result. It has both magnitude and direction, just like any other vector.
The del symbol (∇) represents the "gradient" operator in vector calculus. It is used to calculate the rate of change or the slope of a multivariable function at a particular point.
Del is used in physics to represent the change in a physical quantity with respect to a change in space or time. It is used in various equations, such as the Navier-Stokes equation for fluid flow and the Schrödinger equation for quantum mechanics.
Del is a vector. It is often referred to as a vector operator because it operates on a vector and produces another vector. However, it is important to note that the components of del are not themselves vectors, but rather partial derivatives.
Del is called a vector because it has both magnitude and direction. Although its components are not vectors, they represent the rate of change in a particular direction, making del a vector overall. This concept may seem confusing, but it is a fundamental concept in vector calculus and has been widely accepted in mathematics and physics.