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We first consider the position vector, l:

- where x, y, and z are rectangular unit vectors.

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 3

Since the unit vectors for rectangular coordinates are constants, we have for dl:

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 4

The operator, del: Ñ is defined to be (in rectangular coordinates) as:

- This operator operates as a vector.

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 5

Gradient

- If the del operator, Ñ operates on a scalar function, f(x,y,z), we get the gradient:

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 6

We can interpret this gradient as a vector with the magnitude and direction of the maximum change of the function in space.

- We can relate the gradient to the differential change in the function:

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 7

Since the del operator should be treated as a vector, there are two ways for a vector to multiply another vector:

- dot product and
- cross product.

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 9

Divergence

- We first consider the dot product:
- The divergence of a vector is defined to be:
- This will not necessarily be true for other unit vectors in other coordinate systems.

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 10

To get some idea of what the divergence of a vector is, we consider Gauss' theorem (sometimes called the divergence theorem).

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 11

Gauss' Theorem (Gaub’s Theorem

- We start with:

Surface Areas

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 12

We can see that each term as written in the last expression gives the value of the change in vector A that cuts perpendicular through the surface.

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 13

For instance, consider the first term:

- The first part:
- gives the change in the x-component of A

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 14

The second part,

- gives the yz surface (or x component of the surface, Sx) where we define the direction of the surface vector as that direction that is perpendicular to its surface.

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 15

The other two terms give the change in the component of A that is perpendicular to the xz (Sy) and xy (Sz) surfaces.

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 16

We thus can write:

- where the vector S is the surface area vector.

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 17

Thus we see that the volume integral of the divergence of vector A is equal to the net amount of A that cuts through (or diverges from) the closed surface that surrounds the volume over which the volume integral is taken.

- Hence the name divergence for

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 18

So what?

- Divergence literally means to get farther apart from a line of path, or
- To turn or branch away from.

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 19

Consider the velocity vector of a cyclist not diverted by any thoughts or obstacles:

Goes straight ahead at constant velocity.

(degree of) divergence 0

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 20

Now suppose they turn with a constant velocity

diverges from original direction

(degree of) divergence 0

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 21

Now suppose they turn and speed up.

diverges from original direction

(degree of) divergence >> 0

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 22

Current of water

No divergence from original direction

(degree of) divergence = 0

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 23

Current of water

Divergence from original direction

(degree of) divergence ≠ 0

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 24

+

E-field between two plates of a capacitor.

Divergenceless

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 25

I

b-field inside a solenoid is homogeneous and divergenceless.

divergenceless solenoidal

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 26

+

+

- Two types of vector fields exists:

Electrostatic Field where the field lines are open and there is circulation of the field flux.

Magnetic Field where the field lines are closed and there is circulation of the field flux.

circulation (rotation) 0

circulation (rotation) = 0

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 29

The mathematical concept of circulation involves the curl operator.

- The curl acts on a vector and generates a vector.

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 30

In Cartesian coordinate system:

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 31

Example

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 32

Important identities:

for any scalar function V.

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 33

Stoke’s Theorem

- General mathematical theorem of Vector Analysis:

Closed boundary of that surface.

Any surface

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 34

Given a vector field

- Verify Stoke’s theorem for a segment of a cylindrical surface defined by:

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 35

Note that has only one component:

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 41

The integral of over the specified surface S is

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 42

The surface S is bounded by contour C = abcd.

The direction of C is chosen so that it is compatible with the surface normal by the right hand rule.

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 45

Curl

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 49

curl or rot

- place paddle wheel in a river
- no rotation at the center
- rotation at the edges

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 51

the vector un is out of the screen

- right hand rule
- Ds is surface enclosed within loop
- closed line integral

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 52

Electric Field Lines

Rules for Field Lines

- Electric field lines point to negative charges
- Electric field lines extend away from positive charges
- Equipotential (same voltage) lines are perpendicular to a line tangent of the electric field lines

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 53

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