# Del, divergence, laplacian

I've been reading up on these three recently, and wondered if anyone could confirm what I think they do. I'm not 100% I understand these.

del $$(\bigtriangleup)$$, when applied to a scalar, creates a vector with that scalar as each of the XYZ values. eg

$$\bigtriangleup . x = (x,x,x)$$
$$\bigtriangleup . 3 = (3,3,3)$$

divergence is applied to a vector, and sums the components of the vector into a scalar. eg

$$\bigtriangleup . (x,y,z) = x+y+z$$
$$\bigtriangleup . (1,2,3) = 1+2+3 = 6$$

finally, laplacian. This is the one I'm not as sure about. It's applied to a scalar I think?

$$\bigtriangleup ^2 = \bigtriangleup(\bigtriangleup)$$
$$\bigtriangleup ^2 . x = \bigtriangleup(\bigtriangleup . x)$$
$$= \bigtriangleup((x,x,x))$$
$$= 3x$$

That doesn't seem right (I think I'm meant to end up with a vector). Can laplacian be broken up like that or does it have a special rule?

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kdv
I've been reading up on these three recently, and wondered if anyone could confirm what I think they do. I'm not 100% I understand these.

del $$(\bigtriangleup)$$, when applied to a scalar, creates a vector with that scalar as each of the XYZ values. eg

$$\bigtriangleup . x = (x,x,x)$$
$$\bigtriangleup . 3 = (3,3,3)$$

divergence is applied to a vector, and sums the components of the vector into a scalar. eg

$$\bigtriangleup . (x,y,z) = x+y+z$$
$$\bigtriangleup . (1,2,3) = 1+2+3 = 6$$

finally, laplacian. This is the one I'm not as sure about. It's applied to a scalar I think?

$$\bigtriangleup ^2 = \bigtriangleup(\bigtriangleup)$$
$$\bigtriangleup ^2 . x = \bigtriangleup(\bigtriangleup . x)$$
$$= \bigtriangleup((x,x,x))$$
$$= 3x$$

That doesn't seem right (I think I'm meant to end up with a vector). Can laplacian be broken up like that or does it have a special rule?
No.

Those are differential operators so they must applied to vector fields or scalar fields

The divergence applied to a vector field gives a number .
The gradient applied to a scala field gives a vector.
The laplacian may be applied to a scalar field or to a vector field producing something of the same nature as what it was applied to.

You can't apply those things to a single vector.

ok, is method right? Regardless of the size of the field, are you still doing "that" to each element?

Tom Mattson
Staff Emeritus
Also, I've never seen anyone write the del operator the way that you have done it. It is always written $\nabla$. The Laplacian on the other hand can be written either as $\nabla^2$ or $\bigtriangleup$.