Sketching vector fields

In summary, the conversation discusses how to sketch three different vector fields in the xy-plane. The first vector field, F(r) = 2r, is a 3D vector field and can be simplified to 2xi + 2yj + 2zk. The second vector field, F(r) = -r/||r||3, involves looking at the direction and magnitudes of the vectors and finding symmetries. The third vector field, F(x,y) = 4xi + xj, can be simplified to sqrt(x^2)i + sqrt(y^2)j. The conversation also clarifies the notation of r and |r|.
  • #1
-EquinoX-
564
1

Homework Statement



I am asked to sketch the following vector field in the xy-plane

(a) F(r) = 2r


(b) F(r) = -r/||r||3


(c) F(x,y) = 4xi + xj

Homework Equations





The Attempt at a Solution



Can someone please give me some hints on how to proceed
 
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  • #2
hi EquinoX

first I would look at the direction and magitudes of the vectors & see if you can spot any symmetries

a good way to start is to sketch the vector at several points as well
 
  • #3
yes but for F(r) = 2r this is a 3D vector field, so if I am asked to sketch it in an xy plane, do I just look at the i + j direction? and is 2r basically just 2xi + 2yj + 2zk?

because r is xi + yj + zk?
 
Last edited:
  • #4
Yeah i think that should do

r = sqrt(x^2 + y^2 + z^2) and in the xy plane z = 0, so this shouldn't affect your r anyway
 
  • #5
so r = sqrt(x^2)i + sqrt(y^2)j ?
 
  • #6
sorry bit of confusion over notation
|r| = sqrt(x^2 + y^2 + z^2)

r = xi + yj + zk
 
  • #7
I am asking 2r?
 

What is a vector field?

A vector field is a mathematical concept used to describe the behavior of a vector quantity, such as force or velocity, at every point in space. It is represented visually as a collection of arrows or lines, where the direction and length of each arrow represents the direction and magnitude of the vector at that point.

Why is sketching vector fields important?

Sketching vector fields is important in many fields of science and engineering, as it allows us to visualize and understand the behavior of vector quantities in a given space. It can help us make predictions, analyze data, and solve problems in areas such as fluid dynamics, electromagnetism, and more.

What are some techniques for sketching vector fields?

There are several techniques for sketching vector fields, including using a grid or coordinate system, drawing field lines, and plotting individual vectors. It is also helpful to understand the properties of vector fields, such as flow, divergence, and curl, in order to accurately represent them in a sketch.

What are some common applications of vector fields?

Vector fields have many practical applications in science and engineering. For example, they can be used to model fluid flow in pipes, analyze the electric and magnetic fields around charged particles, and simulate the movement of celestial bodies in space. They are also used in data analysis and image processing.

How can I use sketching vector fields in my research or work?

If you are a scientist or engineer, sketching vector fields can be a useful tool for visualizing and understanding complex systems and phenomena. It can also help you make predictions and test hypotheses in your research. Additionally, many software programs and tools are available to assist with creating accurate and detailed vector field sketches for your work.

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