# Connexions

You are here: Home » Content » Divergence, Gradient, and Curl

### Lenses

What is a lens?

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

#### Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
• Rice Digital Scholarship

This module is included in aLens by: Digital Scholarship at Rice UniversityAs a part of collection: "Waves and Optics"

"This book covers second year Physics at Rice University."

Click the "Rice Digital Scholarship" link to see all content affiliated with them.

Click the tag icon to display tags associated with this content.

### Recently Viewed

This feature requires Javascript to be enabled.

### Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

Module by: Paul Padley. E-mail the author

Summary: A brief introduction to the basic elements of vector calculus.

Assume we have measured the temperature in a room along an axis x x . If we wanted to find the temperature change as we move to postion ( x + Δ x ) ( x + Δ x ) then from the fundamental definition of a derivative we know that is: Δ T = T x Δ x Δ T = T x Δ x

We can easily extend this concept to 3 dimensions At position ( x , y , z ) ( x , y , z ) there is a temperature T ( x , y , z ) T ( x , y , z ) . Suppose we then want to find the temperature at R + Δ R = ( x + Δ x , y + Δ y , z + Δ z ) R + Δ R = ( x + Δ x , y + Δ y , z + Δ z ) . Then we can use: Δ T = T x Δ x + T y Δ y + T z Δ z Δ T = T x Δ x + T y Δ y + T z Δ z

We could define a vector ( T x , T y , T z ) ( T x , T y , T z ) and then say Δ T = ( T x , T y , T z ) Δ R Δ T = ( T x , T y , T z ) Δ R so let's define an operator = ( x , y , z ) = ( x , y , z ) Then we can write Δ T = T Δ R Δ T = T Δ R is a vector operator that can be used in other situations involving scalars and vectors. It is often named "del" or "nabla". Operating on a scalar field with this operator is called taking the "gradient" of the field.

We could also operate on a vector field with del. There are two different ways to do this, by taking the dot and the cross products. To operate on a vector field by taking its dot product with del is referred to as taking the divergence. ie. f = h f = h where h h is some vector field and f f is the resulting scalar field.

Similarly one could take the cross product: g = × h g = × h where g g is the resulting vector field. g x = ( × h ) x = h z y h y z g y = ( × h ) y = h x z h z x g z = ( × h ) z = h y x h x y g x = ( × h ) x = h z y h y z g y = ( × h ) y = h x z h z x g z = ( × h ) z = h y x h x y This is referred to as taking the curl of a field.

These operations, Gradient, Divergence and Curl are of fundamental importance. They have been presented above as operations using some newly defined operator but they in fact have deep physical significance. When using these operators to express Maxwell's equations in differential form, the meaning of these operations will hopefully become more clear. Gradient is the easiest to understand, it can be thought of as a three dimensional slope.

Having defined these operations we can go on to second derivative type things ( T ) = 2 T = a s c a l a r f i e l d ( T ) = 2 T = a s c a l a r f i e l d Note that 2 2 occurs so often that is has its own name, Laplacian × ( T ) = 0 × ( T ) = 0 ( h ) = a v e c t o r f i e l d ( h ) = a v e c t o r f i e l d ( × h ) = 0 ( × h ) = 0 × ( × h ) = ( h ) 2 h × ( × h ) = ( h ) 2 h h = 2 h = a v e c t o r f i e l d h = 2 h = a v e c t o r f i e l d Here is an example of taking a divergence that will be extremely useful. If r = x ı ̂ + y ̂ + z k ̂ r = x ı ̂ + y ̂ + z k ̂ and k = k x ı ̂ + k y ̂ + k z k ̂ k = k x ı ̂ + k y ̂ + k z k ̂ and

E = E x ı ̂ + E y ̂ + E z k ̂ E = E x ı ̂ + E y ̂ + E z k ̂ then lets find (for E E is a constant vector) E e i k r = x E x e i k r + y E y e i k r + z E z e i k r = E x e i k r x ( i k r ) + = i E x e i k r x ( k x x + k y y + k z z ) + = i k x E x e i k r + i k y E y e i k r + i k z E z e i k r = i k E e i k r E e i k r = x E x e i k r + y E y e i k r + z E z e i k r = E x e i k r x ( i k r ) + = i E x e i k r x ( k x x + k y y + k z z ) + = i k x E x e i k r + i k y E y e i k r + i k z E z e i k r = i k E e i k r

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks