- #1

- 459

- 5

Why is it that

[tex]\vec{A}\cdot\nabla \neq \nabla\cdot\vec{A}[/tex]

?

edit: sorry about that. fixed the typo.

[tex]\vec{A}\cdot\nabla \neq \nabla\cdot\vec{A}[/tex]

?

edit: sorry about that. fixed the typo.

Last edited:

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter Swapnil
- Start date

- #1

- 459

- 5

[tex]\vec{A}\cdot\nabla \neq \nabla\cdot\vec{A}[/tex]

?

edit: sorry about that. fixed the typo.

Last edited:

- #2

- 453

- 0

I assume there is some kind of typo in this question?

- #3

- 1,074

- 1

Why is it that

[tex]\vec{A}\cdot\nabla \neq \vec{A}\cdot\nabla[/tex]

?

Did you mean [tex]\vec{A}\cdot\nabla \neq \nabla\cdot\vec{A}[/tex]?

If so then evaluate them both, what do you notice?

- #4

- 13,033

- 588

The first is a differential operator, while the second is a number (value of a function).

- #5

George Jones

Staff Emeritus

Science Advisor

Gold Member

- 7,425

- 1,064

[tex]\vec{A}\cdot\nabla \neq \nabla\cdot\vec{A}[/tex]

?

edit: sorry about that. fixed the typo.

Apply both sides to an arbitary function f = f(x,y,xz). What do you get?

- #6

- 406

- 7

"Dot Del" is what is technically referred to as an "abuse of notation". Of course, some people consider del to be an abuse of notation in itself.

You see, by itself [tex]\nabla[/tex] is just [tex](\frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2}, \ldots , \frac{\partial}{\partial x_n})[/tex]. It's nice but all the derivatives have the same coefficient on them (i.e. one).

To allow for more general operators we use [tex]A \cdot \nabla[/tex] to stand for [tex](a_1 \frac{\partial}{\partial x_1} , a_2 \frac{\partial}{\partial x_2} , \ldots , a_n \frac{\partial}{\partial x_n})[/tex] where [tex]A = (a_1,a_2, \ldots, a_n)[/tex]. It's confusing because usually the dot product is associative, but now we're demanding it no be for [tex]\nabla[/tex].

I'm not a great fan for this notation myself, but since I've got nothing better to offer, I guess I'll just have to live with it.

You see, by itself [tex]\nabla[/tex] is just [tex](\frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2}, \ldots , \frac{\partial}{\partial x_n})[/tex]. It's nice but all the derivatives have the same coefficient on them (i.e. one).

To allow for more general operators we use [tex]A \cdot \nabla[/tex] to stand for [tex](a_1 \frac{\partial}{\partial x_1} , a_2 \frac{\partial}{\partial x_2} , \ldots , a_n \frac{\partial}{\partial x_n})[/tex] where [tex]A = (a_1,a_2, \ldots, a_n)[/tex]. It's confusing because usually the dot product is associative, but now we're demanding it no be for [tex]\nabla[/tex].

I'm not a great fan for this notation myself, but since I've got nothing better to offer, I guess I'll just have to live with it.

Last edited:

- #7

arildno

Science Advisor

Homework Helper

Gold Member

Dearly Missed

- 9,970

- 134

Abuse??

When A has the interpretation of a velocity field, then [itex]\vec{A}\cdot\nabla[/itex] has the very nice interpretation of the convective derivative operator.

The only "abuse" I'm able to see is that the "dot product" is defined for vectors, while the [itex]\nabla[/itex] operator isn't a vector at all.

However, then we must agree that [itex]\nabla\cdot\vec{A}[/itex] is an equally abusive notation.

When A has the interpretation of a velocity field, then [itex]\vec{A}\cdot\nabla[/itex] has the very nice interpretation of the convective derivative operator.

The only "abuse" I'm able to see is that the "dot product" is defined for vectors, while the [itex]\nabla[/itex] operator isn't a vector at all.

However, then we must agree that [itex]\nabla\cdot\vec{A}[/itex] is an equally abusive notation.

Last edited:

- #8

HallsofIvy

Science Advisor

Homework Helper

- 41,833

- 963

"Dot Del" is what is technically referred to as an "abuse of notation". Of course, some people consider del to be an abuse of notation in itself.

You see, by itself [tex]\nabla[/tex] is just [tex](\frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2}, \ldots , \frac{\partial}{\partial x_n})[/tex]. It's nice but all the derivatives have the same coefficient on them (i.e. one).

To allow for more general operators we use [tex]A \cdot \nabla[/tex] to stand for [tex](a_1 \frac{\partial}{\partial x_1} , a_2 \frac{\partial}{\partial x_2} , \ldots , a_n \frac{\partial}{\partial x_n})[/tex]

No, it's not. [itex]A \cdot \nabla[/itex] is

[tex](a_1 \frac{\partial}{\partial x_1} + a_2 \frac{\partial}{\partial x_2} + \ldots + a_n \frac{\partial}{\partial x_n})[/tex]

Share: