All Questions
Tagged with differentiation vectors
120 questions
4
votes
2
answers
2k
views
Confusion with partial derivatives as basis vectors
So I have seen that the directional derivative can be written as
$$ \frac{df}{d\lambda} = \frac{dx^i}{d\lambda}\frac{df}{dx^i} $$
And we can identify $ \frac{d}{dx^i} $ as basis vectors and $ \...
- 4,064
7
votes
4
answers
16k
views
Is Del (or Nabla) an operator or a vector?
Is Del (or Nabla, $\nabla$) an operator or a vector ?
\begin{equation*}
\nabla\equiv\frac{\partial}{\partial x}\vec{i}+\frac{\partial}{\partial y}\vec{j}+\frac{\partial}{\partial z}\vec{k}
\end{...
- 129
2
votes
1
answer
125
views
Vector Derivative: General Case
From "An Introduction to Mechanics" by Kleppner & Kolenkow, SIE-2007, Chapter 1 (Vectors and Kinematics), Section 1.8 - "More about the derivative of a vector".
In this section, towards the end, ...
- 203
1
vote
2
answers
108
views
Two different formulas
My problem is simple : given a particle of mass $m$, charge $q$ and velocity $\bf{v}$. If $\bf{A}$ denotes the magnetic potential satisfying $\bf{B}= \nabla \times \bf{A}$.
I want to etablish the ...
- 123
0
votes
1
answer
100
views
Why trajectories approach to origin tangent to the slower direction?
I am reading non-linear dynamics from Strogartz. Suppose, I have two solutions of a non linear system: $x(t) = x_0e^{at}$ and $y(t) = y_0e^{-t}$, where $a\in \mathbb{R}$. Now it is clear that,for $a&...
- 27
6
votes
4
answers
751
views
Do $\vec r$ and $d \vec r$ have the same direction?
One question is bugging me for a long time but I couldn't make out anything nor could my friends. Here it goes:
We know, $\vec r$ is regarded as the position vector. So we can say,
$$\vec r \cdot\vec ...
- 4,984
4
votes
1
answer
3k
views
Differentiation of a vector with respect to a vector
Does differentiation of a vector with respect to a vector make any sense? Even if it makes sense, how does it make any physical meaning? I mean what is the physical interpretation?
- 4,984
15
votes
5
answers
2k
views
What does it mean for a physical quantity if its mixed second partial derivatives are not equal?
This goes for every problem (either in electromagnetism or fluid dynamics) that has to do with vector fields. Say we have a fluid flowing in a closed circular pipe (or an electromagnetic field, the ...
- 7,860
2
votes
3
answers
179
views
Difference between $|d{\bf r}|$ and $d|{\bf r}|$
What is the difference between $|d{\bf r}|$ and $d|{\bf r}|$ and why are both of them not always equal to each other?
My question might seem stupid to some and will probably get downvoted but I have ...
- 733
2
votes
4
answers
20k
views
How to find tangential/radial/angular velocity for motion in any curve? [closed]
Is the radial velocity responsible only for changing distance between objects and the component perpendicular to it only for change in direction? If so why?
Please try to give a different explanation ...
- 188
-1
votes
3
answers
69
views
Vector question, differentials, Electromagnetism
I was reading this demonstration of electric potential in my book:
Let $q$ be a point charge at point $P$
The Electric field created at point $M$ by $q$ is :
$$\vec{E}(M) = \...
- 685
0
votes
1
answer
268
views
Gradient of two-particle system
I'm working on problem 5.1a from Griffiths Intro to QM and given that:
$$\mathbf R \equiv \frac{m_1\mathbf{r_1} + m_2 \bf r_2}{m_1+m_2}$$
and $\bf r \equiv \bf r_1 - \bf r_2$ I need to show that,
$$\...
- 207
0
votes
1
answer
68
views
The vector r points from $P'(x',y',z')$ to $P(x,y,z)$ [closed]
For some reason this question is giving me a hard time :(
The vector $r$ points from $P'(x',y',z')$ to $P(x,y,z)$.
(a) Show that if $P$ is fixed and $P'$ is allowed to move, then $\nabla'(\frac{1}{r}...
- 817
1
vote
2
answers
4k
views
Is there a difference in handwritten nabla $\vec{\nabla}$ with an overset arrow and typeset nabla $\nabla$?
According to some physicist at KIT it is usual to write the following when using pen and paper:
whereas in typeset texts you write $\nabla$.
Is that true? Are there sources for this convention?
- 955
1
vote
4
answers
9k
views
Dot product of vector and its derivative with respect to time? How does $L \cdot\frac{dL}{dt} = \frac{1}{2}\frac{d(L^2)}{dt}$? [closed]
How does:
$$L \cdot\frac{dL}{dt} = \frac{1}{2}\frac{d(L^2)}{dt}$$
where L is a vector (I dunno how to make it bold in the equation).
How do they reach to this right hand side equation?
And what is ...
- 143
1
vote
1
answer
253
views
Curl of a vector field with two different systems of coordinates
Let
$$\mathbf{H} = H_x \mathbf{u}_x + H_y \mathbf{u}_y + H_z \mathbf{u}_z$$
be a vector field whose components are defined with respect to the unit vectors $\mathbf{u}_x$, $\mathbf{u}_y$ and $\...
- 777
3
votes
1
answer
133
views
Contradiction of a scalar product
Can anyone resolve this contradiction:
$$\vec{r}\cdot\dot{\vec{r}}=\frac{1}{2}\frac{d}{dt}\left(\vec{r}^2\right)=\frac{1}{2}\frac{d}{dt}\left(\left|\vec{r}\right|^2\right)\equiv\frac{1}{2}\frac{d}{dt}...
- 393
0
votes
2
answers
5k
views
How is the direction of the instantaneous acceleration determined?
I know from the text book that the direction of velocity at any point on the 2D path of an object is tangential to the path at that point and is in the direction of motion. But how would one determine ...
- 941
15
votes
3
answers
44k
views
Derive vector gradient in spherical coordinates from first principles
Trying to understand where the $\frac{1}{r sin(\theta)}$ and $1/r$ bits come in the definition of gradient.
I've derived the spherical unit vectors but now I don't understand how to transform ...
- 498
7
votes
6
answers
8k
views
How is gradient the maximum rate of change of a function?
Recently I read a book which described about gradient. It says
$${\rm d}T~=~ \nabla T \cdot {\rm d}{\bf r},$$
and suddenly they concluded that $\nabla T$ is the maximum rate of change of $f(T)$ ...
- 872