What is Vector calculus: Definition and 419 Discussions
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space
R
3
.
{\displaystyle \mathbb {R} ^{3}.}
The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of
electromagnetic fields, gravitational fields, and fluid flow.
Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, Vector Analysis. In the conventional form using cross products, vector calculus does not generalize to higher dimensions, while the alternative approach of geometric algebra which uses exterior products does (see § Generalizations below for more).
Vector Calculus
Greetings everyone,
I have a problem: "Consider the curves r1(t)=(t, 1-t, 16+t^2) and
r2(t)=(8-s,s-7, s^2)
a) At what point do they meet?
b) Find their angle of intersection
The first part is easy, but I'm encountering some problems with b). To find the angle we need...
the problem goes:
ABCD is a parallelogram in which points P and Q are the midpoints of sides BC and CD, respectively. Use vector calculus to prove that AP and AQ trisect the diagonal BD at the points E and F.
...A _________B
.../...F.../
.../...E.../P
D/________/C
...Q
(Imagine...
Ok here is the problem:
Given two lines in space, either they are parallel, or they intersect or they are skew. Determine whether the lines taken two at a time, are parallel, intersect or are skew. If they intersect find the point of intersection.
line 1: x=1+2t, y=-1-t, z=3t...
i having so much trouble with this vector calculus question, someone please help.
At time t, a moving particle has position r(t)=((e^t) + (e^-t))/2 i + ((e^t) - (e^-t))/2 j.
a) find the cartesian equation of the path
I know all you have to do is let the i component to equal x(t) and...
Could anyone tell me why Vector Calculus would be a prequisite or a co-requisite? Specifically, what topics are required to know vector calculus in PDE's?
I suspected a course in ODE's would be enough.
Thanks
A vector field is defined by \vec{A}(\vec{r})} = \rho^2 \hat{\phi} + \rho \sin \phi \hat{z}
Verify Stokes' theorem by explicit calcluation where S is the circle of radius a in the z = 0 plane.
You may like to employ the ideas that \vec{dS} = \rho d\phi dz \hat{\rho} + dz d\rho \hat{\phi} +...
In my physics book, the 4 properties of an ideal gase are
1. nonviscous
2. steady flow (laminar)
3. incompressible
4. irrotational
My question is the properties of being irrotional the same as the vector functions that have a Curl=O iff irrotational
My physics book states the...
I'm doing an assignment where the lecturer has said scalar function g(x,y,z) = x^3 + y + z^2
and vector field F = (2xz,sin y,e^y)
and asked find
a) grad g
which is fairly easy, but then
b) div g
and my understanding was that you can only find the divergence of a vector...
Hi All,
I need some suggestion on a good book for vector calculus/advanced vector calculus.
current book I am reading just give equations like
del x ( A x B ) = A del.B - Bdel.A + (B.del)A - (A.del)B
A x ( B x C ) = B(del.A) - C(A.B)
del x (f A) = f del x A + del f x A
etc...
Anyone take a look at this vector calculus question for me:
=====
Q. If n is the uni normal to the surface S, evaluate Double Integral r.n dS over the surface of a sphere of radius 'a' centred at the origin.
=====
So I did:
r = (x,y,z)
Sphere: x^2 + y^2 + z^2 = a^2
let f = x^2 +...
Hi, I have this vector calculus question to do, and I can't seem to get it right! Could someone take a look for me?
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Q. The vector A(r) = (y,-x,z). Verify Stokes' Theorem for the hemispherical surface |r|=1, z>=0.
A. I considered, the line integral about the circle in the xy plane...
f = 2xy in the x dir, (x^2 - z^2) in the y dir, -3xz^2 in the z dir.
particle travles from (0,0,0) to (2,1,3)
im supposed to find the line integral from (0,0,0) to (2,1,3).
if i make z = 3/2 x and y = 1/2 x and substitute those into the integral of F dot dl i end up with an answer of...
f = 2xy in the x dir, (x^2 - z^2) in the y dir, -3xz^2 in the z dir.
particle travles from (0,0,0) to (2,1,3) along the segments (0,0,0) ->(0,1,0)->(2,1,0)->(2,1,3)
the integral is F dot product with dl.
i can't figure this out.
do i dot product each part individually and evaluate? i...
I have a problem that I don't even know how to start. Can anyone assist me with the following problem?
The tension T at the end of each chain has magnitude 25 N and makes an angle of 37 degrees with the horizontal. What is the weight of the chain?
Thanks in advance.
I'm a week or two away fromt the end of my vector calculus class and we are covering topics like surface integration, green's theorem and such. The problem I'm having so far is that everything just seems so disjointed and ad hoc, with all these theorems I feel I have to memorize instead of...
Hi All,
I've been working through a series of vector calculus problems and I need some help to get started with this one. Anyone care to help?
The problem is:
Evaluate \oint_{s} r.n.dS where S is a closed surface.
Cheers
Vector Calculus Proof Help Please :)
Heya Ppl i have a problem i am trying to solve.
Prove that
(Delta) . ( (fi)F) = (fi)(Delta) . F + F . (Delta)(fi)
were these contain GRAD DIV in my opinion but i seem to not be able to get the answer.
F = Vector F where F = F1i + F2j + F3k is a...
Hi All,
I've started to study vector calculus and am finding it a bit daunting. The notes the lecturer has supplied aren't too helpful. Does anyone know of some decent resources on the web that could help me out?
Many Thanks,
Pete
Can anyone give me a very brief summary of what vector calculus means? I know this sounds like a "specify what you mean" type question, but I hope it isn't. Let me explain further. I know all the equations, how to find line integrals, what Green's theorem is, etc. but I don't exactly know what...