What is Vector calculus: Definition and 422 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).
##\vec{E}## on the line that perpendicularly bisects the segment that joins two equal and opposite charges is non-zero, as it should be. But the potential of any point along that line is zero. But we know know that ##E=-\frac{dV}{dr} ##, where V is approximately ##\frac{1}{4\pi \epsilon}...
I have seen conservative vector fields being defined as satisfying either of the two following conditions:
The line integral of the vector field around a closed loop is zero.
The line integral of the vector field along a path is the function of the endpoints of the curve.
It is apparent to me...
Hi, a doubt about the definition of vector space.
Take for instance the set of polynomals defined on a field ##\mathbb R ## or ##\mathbb C##. One can define the sum of them and the product for a scalar, and check the axioms of vector space are actually fullfilled.
Now the point is: if one...
Hi,
I'm aware of the ##L^2## space of square integrable functions is an Hilbert space.
I believe the condition to be ##L^2## square-integrable actually refers to the notion of Lebesgue integral, i.e. a measurable space ##(X,\Sigma)## is tacitly understood. Using properties of Lebesgue integral...
Attempt : (Turns out, there is more mathematics in this problem than physics. The crucial part involves the use of vector calculus where one needs to find the volume of a region bounded at the top by a portion of a sphere. That is where am stuck.)
The mass of water displaced by the ball...
Solving the integral is the easiest part. Using spherical coordinates:
$$ \oint_{s} \frac{1}{|\vec{r}-\vec{r'}|}da' = \int_{0}^{\pi}\int_{0}^{2\pi} \frac{1}{|\vec{r}-\vec{r'}|}r_{0}^2 \hat r \sin{\theta}d\theta d\phi$$
then:
$$I = \dfrac{1}{|\vec{r}-\vec{r'}|}r_{0}^2(1+1)(2\pi)\hat...
So, curl of curl of a vector field is, $$\nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$$
Now, curl means how much a vector field rotates counterclockwise. Then, curl of curl should mean how much the curl rotate counterclockwise.
The laplacian...
During the calculations, I tried to solve the following
$$ \vec{\nabla} \big[\vec{M}\cdot\vec{\nabla} \big(\frac{1}{r}\big)\big] = -\big[\vec{\nabla}(\vec{M}\cdot \vec{r}) \frac{1}{r^3} + (\vec{M}\cdot \vec{r}) \big(\vec{\nabla} \frac{1}{r^3}\big) \big]$$
by solving the first term i.e...
As you can see in the homework statement, I am asked to calculate what's effectively the divergence of the vector field ##\vec{F} = \vec{x}/\vert\vec{x}\vert^3## over ##\mathbb{R}^3##. I have done that, the calculation itself isn't that difficult after all. However, I can't make sense of the...
In Feymann's seminar on superconductivity, there was this equation (21.28) ##\oint_C \nabla \theta\cdot dl = \frac q \hbar \Phi##. But the gradient theorem demands that ##\oint_C \nabla \theta\cdot dl=0##
We were taught that in cylindrical coodrinates, the position vector can be expressed as
And then we can write the line element by differentiating to get
.
We can then use this to do a line integral with a vector field along any path. And this seems to be what is done on all questions I've...
[mentor's note - moved from one of the homework help forums]
Homework Statement:: It's a question.
Relevant Equations:: Vector calculus.
Is it true to say that in one dimension I can show vector quantities using ±number instead of a vector?
± can show possible directions in one dimension and...
In physics there is a notation ##\nabla_i U## to refer to the gradient of the scalar function ##U## with respect to the coordinates of the ##i##-th particle, or whatever the case may be.
A question asks me to prove that
$$\nabla_1U(\mathbf{r}_1- \mathbf{r}_2 )=-\nabla_2U(\mathbf{r}_1-...
This is probably a stupid question, but I have never realised that there's an order things should be done in the chain rule , so for example
## \nabla(\bf{v}.\bf{v})=2\bf{v} (\nabla\cdot \bf{v}) ##
and not
## 2 \bf{v} \cdot \nabla \bf{v} ##
Is there an obvious way to see / think of this...
TL;DR Summary: My problems comes to a vector expression which needs to be simplified
I got an expression
pi=εijksk,lul,j
Here s and u are two vectors. What will be the vector expression of this vector p with curl s, curl u, and other operations?
##f\left(x\right)=\begin{cases}\sqrt{\left|xy\right|}sin\left(\frac{1}{xy}\right)&xy\ne 0\\ 0&xy=0\end{cases}##
I showed it partial derivatives exist at ##(0,0)##, also it is continuous as ##(0,0)##
but now I have to show if it differentiable or not at ##(0,0)##.
According to answers it is not...
I don't have any idea about how to use the hint given by the author.
Author has given the answer to this question i-e F(x,y) = axy + bx + cy +d.
I don't understand how did the author compute this answer.
Would any member of Physics Forums enlighten me in this regard?
Any math help will be...
Homework Statement:: Find the instantaneous acceleration of a projectile fired along a line of longitude (with angular velocity of ##\gamma##constant relative to the sphere) if the sphere
is rotating with angular velocity ##\omega##.
Relevant Equations:: None
Find the instantaneous...
Summary: Consider a body which is rotating with constant angular velocity ω about some
axis passing through the origin. Assume the origin is fixed, and that we are sitting
in a fixed coordinate system ##O_{xyz}##
If ##\rho## is a vector of constant magnitude and constant direction in the...
Summary: Evaluate ##\displaystyle\iint\limits_R e^{\frac{x-y}{x+y}} dA ## where ##R {(x,y): x \geq 0, y \geq 0, x+y \leq 1}##
Author has given the answer to this question as ## \frac{e^2 -1}{4e} =0.587600596824 ## But hp 50g pc emulator gave the answer after more than 11 minutes of time...
Hi.
I have the Marsden an Tromba vector calculus book 6th edition.
I was wondering which software was used to create the books graphs.
I attach two graphs as an example.
Thanks
Hi, Hi,
Author said If we look at the graph of $ f (x, y)= (x^2 +y^2)*e^{-(x^2+y^2)},$ as shown in the following Figure it looks like we might have a local maximum for (x, y) on the unit circle $ x^2 + y^2 = 1.$
But when I read this graph, I couldn't guess that the stated function have a...
In Zettili book, it is given that ## \nabla^2 \psi \left( \vec{r} \right) + \dfrac{1}{\hbar ^2} p^2 \left( \vec{r} \right) \psi ( \vec{r} ) =0 ## where ## \hbar## is very small and ##p## is classical momentum.
Now they assumed the ansatz that ## \psi ( \vec{r} ) = A ( \vec{r} ) e^{i S( \vec{r} )...
I have a few questions about the negative Bendixon criterion. In order to present my doubts, I organize this post as follows. First, I present the theorem and its interpretation. Second, I present a worked example and my doubts.
The Bendixson criterion is a theorem that permits one to establish...
I'm not sure where to start, when I tired using integration of the initial equation to get pos(t)=-.65t^2 i + .13t^2 j + 14ti +13tj but after separating each component, i and j, and setting j equal to zero I got 0 or -100 seconds which doesn't seem like a reasonable answer.
This is the question,
Now to my question, supposing the vectors were not given, can we let ##V=\vec {RQ}## and ##W=\vec {RP}##? i tried using this and i was not getting the required area. Thanks...
The curl is defined using Cartersian coordinates as
\begin{equation}
\nabla\times A =
\begin{vmatrix}
\hat{x} & \hat{y} & \hat{z} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
A_x & A_y & A_z
\end{vmatrix}.
\end{equation}
However, what are the...
Consider the following
\begin{equation}
\nabla\phi=\nabla\times \vec{A}.
\end{equation}
Is it possible to find ##\vec{A}## from the above equation and if so, how does one go about doing so?
[Moderator's note: moved from a homework forum.]
I saw this in a textbook and I thought it is a corollary of Reynold's transport theorem. Let \mathbf{F} be a smooth vector field Consider the surface integral:
\int_{S}\mathbf{F}\cdot d\mathbf{S} and now take the derivative of it, then the expression can be written as...
Hello,
Can someone explain how to sketch the flow profile in detail. Also, I solved for curl, but I'm getting a zero while the answer is the differentiation of the function f(y). Pls do help me out!
Given surface ##S## in ##\mathbb{R}^3##:
$$
z = 5-x^2-y^2, 1<z<4
$$
For a vector field ##\mathbf{A} = (3y, -xz, yz^2)##. I'm trying to calculate the surface flux of the curl of the vector field ##\int \nabla \times \mathbf{A} \cdot d\mathbf{S}##. By Stokes's theorem, this should be equal the...
Since the question asks for Cartesian coordinates, I wrote dV as 2pi(x^2+y^2+z^2)dxdydz and did the integral over the left hand side of the equation with x, y, z from 0 to R. My integral returned (0, 2*pi*R^5, 5/3*pi*R^6) which doesn't seem right.
I also tried to compute the right-hand side of...
Majoring in electrical engineering imply studying Griffiths book on electrodynamics, so I have begun reading its first chapter, which is a review of vector calculus. A list of vector calculus identities is given, and I would like to derive each one, with one of them being ##\nabla \cdot (A...
Hi,
In $\mathbb{R^3} || v-w ||^2=||v||^2 + ||w||^2 - 2||v||\cdot ||w||\cos{\theta}$ But can we say $||v+w||^2=||v||^2 +||w||^2 + 2||v|| \cdot||w|| \cos{\theta}$ where v and w are any two vectors in $\mathbb{R}^3$
If say we have a scalar function ##T(x,y,z)## (say the temperature in a room). then the rate at which T changes in a particular direction is given by the above equation)
say You move in the ##Y##direction then ##T## does not change in the ##x## and ##z## directions hence ##dT = \frac{\partial...
Here is how my teacher solved this:
I understand what the nabla operator does, ##∇\cdot\vec v## means that I am supposed to calculate ##\sum_{n=1}^3\frac {d\vec v} {dx_n}## where ##x_n## are cylindrical coordinates and ##\vec e_3 = \vec e_z##. I understand why ##∇\cdot\vec v = 0##, I would get...
##(\nabla\times\vec B) \times \vec B=\nabla \cdot (\vec B\vec B -\frac 1 2B^2\mathcal I)-(\nabla \cdot \vec B)\vec B##
##\mathcal I## is the unit tensor
##\vec F= 2x^2y \hat i - y^2 \hat j + 4xz^2 \hat k ##
## \Rightarrow \vec \nabla \cdot \vec F= 4xy-2y+8xz##
Let's shift to a rotated cylindrical system with axis on x axis:
##x \to h, y \to \rho cos \phi, z \to \rho sin \phi ##
Then our flux, as given by the Divergence theorem is the volume...
Given $\vec{r}=t^m* \vec{A} +t^n*\vec{B}$ where $\vec{A}$ and $\vec{B}$ are constant vectors,
How to show that if $\vec{r}$ and $\frac{d^2\vec{r}}{dt^2}$ are parallel vectors , then m+n=1, unless m=n?
I don't have any idea to answer this question. If any member knows the answer to this...
Hi,
Let f(t) be a differentiable curve such that $f(t)\not= 0$ for all t. How to show that $\frac{d}{dt}\left(\frac{f(t)}{||f(t)||}\right)=\frac{f(t)\times(f'(t)\times f(t))}{||f(t)||^3}\tag{1}$
My attempt...
In this image of Introduction to Electrodynamics by Griffiths
.
we have calculated the vector potential as ##\mathbf A = \frac{\mu_0 ~n~I}{2}s \hat{\phi}##. I tried taking its curl but didn't get ##\mathbf B = \mu_0~n~I \hat{z}##. In this thread, I have calculated it like this ...
I have a vector field which is originallly written as $$ \mathbf A = \frac{\mu_0~n~I~r}{2} ~\hat \phi$$ and I translated it like this $$\mathbf A = 0 ~\hat{r},~~ \frac{\mu_0 ~n~I~r}{2} ~\hat{\phi} , ~~0 ~\hat{\theta}$$(##r## is the distance from origin, ##\phi## is azimuthal angle and ##\theta##...