vector field

One is to say that the vector field is a gradient in a certain region of a plane.

This is just a particular kind of function that you get out of a vector field, but it is like any function.

It measures how much a vector field goes across the curve.

Here is the statement. The curl is just going to measure how much your vector field fails to be conservative.

What changes is, of course, my vector field is no longer sticking straight out so I cannot use this easy geometric argument.

Well, I want to figure out how much my vector field is going across that surface.

That would be an example of a vector field that comes up in physics.

That means you need your vector field to be continuous and differentiable everywhere on the surface S.

And so now, remember, curl of a vector field in space is going to be another vector expression. It has three components.

Let's not worry too much about it. For accuracy we need our vector field to be defined in a simply connected region.

So, let's look a little bit about one kind of vector field that actually we've encountered a few weeks ago without saying it.

The problem comes from a vector field satisfying this criterion in a region but it has a hole in it.

So, I said, if there is a point somewhere in here where my vector field is not defined, then it doesn't work.

And, of course, it is not a coincidence because this vector field is a gradient field.

Remember that was the vector field that looked like a rotation at the unit speed.

It measures failure of a vector field to be conservative.

The electric current density vector field also satisfies superposition principle, some applications thereof are introduced.

Well, if I take the vector field that was in the problem set, and if I do things, say that I look at the unit circle.

So, we need to set up the flux integral for a vector field dot ndS.

OK, we have to look at the flux of this vector field through the entire surface, S, which is the whole boundary of D.

It's a vector field that just rotates around the origin counterclockwise.

So, if the gradient of a function is a vector, the divergence of a vector field is a function.

Well, I will just prove the same thing for a vector field that has only an x component or only a y component.

An enhanced streamline algorithm is presented. It generates high-quality textures to represent the vector field on surfaces.

Remember, the divergence of a vector field What do these two theorems say?

Well, it is the stuff that lives in a small box whose base is that piece of surface and whose other side is given by the vector field.

So, to say that a vector field with conservative means 0 that the line integral is zero along any closed curve.

If I'm on the x-axis, my vector field is actually in the y direction.

What you do is just-- R Say that your vector field has components P, Q and R.

OK, so just to reinforce my warning that not every field is a gradient field, let's look again at our favorite vector field from yesterday.

The eigenvalue equation and approximate formula for cut-off frequencies of guided modes are obtained by vector-field-method.

But it's a new vector field that you can build out of f.

Actually, they have a vector field is still pointing perpendicular to the level curves that we have seen, just to remind you.

It is well known that the sound wave has both scalar quantity and vector field, and pressure is scalar, particle velocity is vector.

Now you compare f sub y as given to you by the vector field with the formula you get from this expression for f.

The curl of a vector field in space is actually a vector field, not a scalar function. I have delayed the inevitable.

And my vector field represents how things are moving at every point of the plane.

The thing to check is whether the curl of our vector field is zero.

Could it be that, by accident, that vector field was a gradient field?

The results showed that the distribution of the precipitable water field was different from the vector field of water vapor flux .

If we have a vector field that cannot possibly be a gradient then we shouldn't try to look for a potential.

A three-dimensional vector field measurement method is demonstrated using the weak magnetic field fiber optic sensor.

That vector field is actually defined everywhere except on the z axis, and it violates the usual theorem that we would expect.

Now, an important difference between curl here and curl in the plane is that now the curl of a vector field is again a vector field.

Well, it looks like positive should win because here the vector field is much larger than over there.

OK, so you take the divergence of a vector field.

And we used a vector field that represents the flow of whatever the substance is whose diffusion we are studying.

If you remember what we did several times, well, along the circle that vector field now is tangent to the circle.

We dot our favorite vector field with it.

Let's say that we had the same C, but now the vector field .

What we will do is just, at every point along the curve, the dot product between the vector field and the normal vector.

So, it's not completely obvious what it looks like, but here is a computer plot of that vector field.

Just to remind you, a vector field in space is just the same thing as in the plane.

If you take the gradient of this, you should get again this vector field over there.

We have three conditions, F= so our criterion -- Vector field F equals .

And we looked at the component of a vector field in the direction that was normal to the curve.

And, this guy is a function that somehow relates to the vector field.

Say that F, our vector field is actually the gradient of some function.

So, this vector field is not conservative.

The problem is not every vector field is a gradient.

Let's say that our vector field has two components.

We present a novel approach for vector field guided solid texture synthesis from 2D examples.

OK, so the first property that I will have for a vector field is that it's conservative.

Vector Blur uses maps to define a vector field along which the blur will take place.

But that assumes that your vector field is well-defined there.

But, you can also use it to define a vector field in space just with no z component.

But, if you have, so, again, the argument would be, well, if a vector field is defined on the curve, it's also defined inside.

Let's say that I have a plane curve and a vector field in the plane.

Gravitational field is a kind of vector field, and the description of gravitational force by general relativity is very effective.

OK, so that's the special case where N is zero, where you have only an x component for your vector field.

It is proportional to this vector field.

That is called the curl of a vector field.

Flux is integral F dot n ds, and that one corresponds to summing the normal component of a vector field.

And the first such place is when we try to test whether a vector field is a gradient field.

That was a vector field in the plane.

s Let's say that we have a vector field and s, a surface in space.

Which k hat? Oh, you mean the vector field.

One is the vector field whose flux you are taking.

In fact, our vector field and our normal vector are parallel to each other.

We had a curve in the plane and we had a vector field.

There are various ways to define the vector field.

Let's say that I give you this vector field. And this a here is a number.

I need to have a vector field that is defined and differentiable -- -- everywhere in D, so same instructions as usual.

OK, so let's assume that we have a vector field whose curl is zero.

So, it's an example of a vector field that is not conservative.

You have seen that in the plane it is already pretty hard to draw a vector field.

Let's see. The vector field will be yellow today.

We have a vector field that gives us a vector at every point.

We need, actually, a vector field that is well-defined everywhere.

OK, so in particular, if you have a vector field that's defined everywhere the plane, then you take any closed curve.

I have a curve in the plane and I have a vector field.

I will get zero. OK, so this is actually how you prove that 0 if your vector field has curve zero, then it's conservative.

That actually is what we will call later a vector field.

We have a sphere of radius a. I have my vector field.

I want to find the potential for this vector field.

And we don't know what little f is, but we know what the vector field is so we can compute that.

Well, actually here it is not very hard to find a function whose gradient is this vector field.

Uniqueness of the singular point of a vector field on Riemannian manifold and its simple Newton's iteration

Imperfect Bifurcation of One-Dimensional Vector Field with a Time-Dependent Parameter

Sub-regional Descriptions of Global GPS Vector Field and Its Geodynamic Significance

The simple Newton's iteration of a singular vector field on Riemannian manifold

Key Technologies and Applications of Vector Field Interactive Visualization in VR Environment

Vector Field Decomposition Approach to Extracting and Visualizing Vortex Features in Planar Flows

Scalar-Field Model for Dark Energy with a Fixed Background Vector Field

Iris Recognition by Restructuring Characteristic Vector Field

Rotate Degress of Vector Field Determines Non-Existence of Closed Orbit about Plane(Space) Differential System

Classification of phase portrait about planar quintic Hamiltonian vector field

The Position Vector Field of a Special Surface about Given Principal Curvature Function

An approximate solution of the Gaussian beam in the vector field and radiation pressure of the beam

Some Results about Functional, Operator, Vector Field in Rn Related with White Noise Functional

An Algorithm of Mapping Additional Scalar Value to Imaged Based Vector Field Visualization

To Define the Relation between Contour Surface and Gradient in Vector Field through Experiment

Directional Derivative of the Vector Field and Regular Curves on Time Scales

A Analysis on the Bifurcation of the Planar Vector Field Which Has the Double--Zero Eigenvalues

A Fast Motion Estimation Algorithm Using Adaptive Motion Vector Field Search

Tennis Video Analysis Using Transformed Motion Vector Field

Separation of Longitude and Shear Wave in three-dimensional Vector Field

Optimization Analysis of Path Planning of Micro Soccer Robot in Vector Field

divergence (of a vector field)

An Improved LIC Algorithm for Vector Field Visualization

For the vector field, we use a trace-form model for its processing after analyzing how to choose certain parameters in the model;

A LIC Method for Visualization of Unstructured Vector Field

Study on Display Technique of Vector Field in Mold Filling Process

An Application of Rotation of Continuous Vector Field

A Fast Motion Estimation Algorithm Based on Motion Vector Field and Direction Adaptive Techniques

New vector field microcosmic model for traffic flow

scalar flow of vector field

vector flow of vector field

physical coordinates of vector field

Rotation of a vector field

Vector Field-oriented Control for Doubly-fed Motor

A New Planar Vector Field Visualization Method

Study of Image Mosaic Based on Oriented Vector Field

Moving object detection and tracking in video sequences based on boundary vector field;

Circulation of the vector field

Vector Field Visualization Based on Texture Synthesis

An Analogue Rotated Vector Field of Polynomial System

A New Adaptive Motion Search with Elastic Pattern Based on Motion Vector Field