New Post(Jan 2008)
The section “The basic idea” appears unnecessarily complicated: Scalar is not defined in a simple manner, neither is vector. Neither are particularly complex conceptions at this level. Either the assumption is that the words require no explanation, which is unlikely in view of the next paragraph, or they do require one, in which case, the concept is, as I say below, not difficult.
There is the sentence, “Detailed explanation of vectors may be found at Wikibooks linear algebra.” Would this be necessary were the article to be clearer? Indeed the detailed explanation may not be needed at all until later in the course when a little more groundwork has been covered.)
“A unit vector is a special vector which has magnitude 1 and points along one of the coordinate frame's axes*. This is better illustrated by a diagram.” – no diagram – but the explanation here is clear enough. (*coordinate frame's = graph's)
Simply put, a scalar is a number on the axis of a graph; in 2x + 3y – 8z • the 2, the 3 and the -8 are scalar – they tell you the quantity of the units on the axes. • The x and the y are the axes of the graph and tell you what the thing is (time / size / quantity / quality / etc) and • a vector is the line (usually, but not always, starting at 0) that is resultant of the other lines and ends at the point indicated by the numbers (scalars) on the axes. (see diagrams.) The size and direction of the vector is important and is usually what is being sought
Thus the vector 2x + 3y +4z is the line to the point at which right angled lines from 2 on the x axis, 3 on the y axis and 4 on the z axis intersect. This is the vector 2x + 3y +4z
The final line of this section reads:
“The magnitude of a vector is computed by (root of sum equation). For example, in two-dimensional space, this equation reduces to (root (x squared plus y squared)) .”
This is a masterpiece of obscurity. 1. “The magnitude of a vector is computed by...” I’m sure, should read, “You can find the magnitude of a vector with this formula…” 2. But in the formula, (root of sum) why is it not clear what x and i are? 3. The term, “in two-dimensional space” can be replaced with, “if there are only 2 axes” 4. There is no explanation or even a hint as to why (root of sum) “reduces to” (root (x squared plus y squared)) and 5. the final equation (root (x squared plus y squared)) is no more than Pythagoras’s Theorem, although heavily (and to my mind pointlessly) disguised.
Let me say that the idea of Wikiversity is excellent and I am, despite the above, grateful to and impressed by the person who gave his/her time to this section. He/she clearly knows the subject like the back of their hand; however, many would-be readers (and I am living proof) have not had his/her years to stare at the back of his/her hand.
- To both of the previous commentators: Thank you very much for your feedback.
- Indeed Wikiversity is a work in progress, we are a wiki - this means just be bold to optimize the text. Here you find more info about Wikiversity: Template:Welcome. If you have more questions, you can also visit the Wikiversity:Chat, ----Erkan Yilmaz (Wikiversity:Chat, wiki blog) 07:35, 15 January 2008 (UTC)
- “A unit vector is a special vector which has magnitude 1 and points along one of the coordinate frame's axes*. This is better illustrated by a diagram.” – no diagram – but the explanation here is clear enough. (*coordinate frame's = graph's)
- 1 Error in text (not a typo)
- 2 moved a large section to another page
- 2.1 BEGIN TEXT THAT HAS BEEN MOVED TO Vector calculus
- 2.2 Derivative of a vector valued function
- 2.3 Scalar and vector fields
- 2.4 Gradient of a scalar field
- 2.5 Divergence of a vector field
- 2.6 Curl of a vector field
- 2.7 Laplacian of a scalar or vector field
- 2.8 Identities in vector calculus
- 2.9 Green-Gauss Divergence Theorem
- 2.10 Stokes' theorem
- 3 Free and fixed vectors?
Error in text (not a typo)
This is wrong:
If you just look at the figure, you can see that the component of a vector along the basis vector is NOT equal to it's component, . You need to establish a dual space such as the reciprocal lattice basis in solid state physics.
You could also represent the same vector in terms of another set of basis vectors () as shown in Figure 1(b). In that case, the components of the vector are and we can write
Note that the basis vectors and do not necessarily have to be unit vectors. All we need is that they be linearly independent, that is, it should not be possible for us to represent one solely in terms of the others.
moved a large section to another page
I moved a large part of this page to Vector calculus
This what was moved:
BEGIN TEXT THAT HAS BEEN MOVED TO Vector calculus
So far we have dealt with constant vectors. It also helps if the vectors are allowed to vary in space. Then we can define derivatives and integrals and deal with vector fields. Some basic ideas of vector calculus are discussed below.
Derivative of a vector valued function
Let be a vector function that can be represented as
where is a scalar.
Then the derivative of with respect to is
Note: In the above equation, the unit vectors (i=1,2,3) are assumed constant.
If and are two vector functions, then from the chain rule we get
Scalar and vector fields
Let be the position vector of any point in space. Suppose that there is a scalar function () that assigns a value to each point in space. Then
represents a scalar field. An example of a scalar field is the temperature. See Figure4(a).
If there is a vector function () that assigns a vector to each point in space, then
represents a vector field. An example is the displacement field. See Figure 4(b).
Gradient of a scalar field
Let be a scalar function. Assume that the partial derivatives of the function are continuous in some region of space. If the point has coordinates () with respect to the basis (), the gradient of is defined as
In index notation,
The gradient is obviously a vector and has a direction. We can think of the gradient at a point being the vector perpendicular to the level contour at that point.
It is often useful to think of the symbol as an operator of the form
Divergence of a vector field
If we form a scalar product of a vector field with the operator, we get a scalar quantity called the divergence of the vector field. Thus,
In index notation,
If , then is called a divergence-free field.
The physical significance of the divergence of a vector field is the rate at which some density exits a given region of space. In the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into or out of the region.
Curl of a vector field
The curl of a vector field is a vector defined as
The physical significance of the curl of a vector field is the amount of rotation or angular momentum of the contents of a region of space.
Laplacian of a scalar or vector field
The Laplacian of a scalar field is a scalar defined as
The Laplacian of a vector field is a vector defined as
Identities in vector calculus
Some frequently used identities from vector calculus are listed below.
Green-Gauss Divergence Theorem
Let be a continuous and differentiable vector field on a body with boundary . The divergence theorem states that
where is the outward unit normal to the surface (see Figure 5).
In index notation,
According to Wikipedia, this form of the theorem was first discovered by Lord Kelvin, who communicated it to George Stokes in a letter dated July 2, 1850. Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name.
END OF TEXT THAT WAS MOVED
Free and fixed vectors?
I'm not sure we need to disinguish between free vectors and fixed vectors. I appreciate motive for making the distinction, but vectors are generally defined to be free.---Guy vandegrift (discuss • contribs) 10:27, 6 April 2016 (UTC)