Doing R=|r-r'|, i get the expected result: \nabla \frac{1}{|r-r'|} = -\frac{1}{R^2}\hat r=-\frac{(r-r')}{|r-r'|^3}
But doing it this way seems extremely wrong, as I seem to be disregarding the module. So I tried to do it by the chain rule, and I got:
\nabla...
Definition: Let f be a differentiable real-valued function on ##\mathbf{R}^3##, and let ##\mathbf{v}_P## be a tangent vector to it. Then the following number is the derivative of a function w.r.t. the tangent vector
$$\mathbf{v}_p[\mathit{f}]=\frac{d}{dt} \big( \mathit{f}(\mathbf{P}+ t...
Assuming that both the Earth and Mars's atmospheric pressure follows an exponential curve, how many kilometers deep would the average bore-hole on Mars need to be in order to arrive at a depth where the atmospheric pressure was 0.35 bar or approximately 5 psi? What about 0.7 bar?
Is there a limit to how steep a refractive index gradient can be before ray optics are no longer able to predict the path of the light? How is it related to wavelength? Under what conditions the light will be able to travel perpendicular to the gradient
In a straight line? (having diffrent index...
I have seen two main different methods for finding the gradient of a vector from various websites but I'm not sure which one I should use or if the two are equivalent...
The first method involves multiplying the gradient vector (del) by the vector in question to form a matrix. I believe the...
Homework Statement
The vlasov equation is (from !Introduction to Plasma Physics and Controlled Fusion! by Francis Chen):
$$\frac{d}{dt}f + \vec{v} \cdot \nabla f + \vec{a} \cdot \nabla_v f = 0$$
Where $$\nabla_v$$ is the del operator in velocity space. I've read that $$\nabla_v =...
Hi Folks,
Was just curious as to what is the gradient of a divergence is and is it always equal to the zero vector. I am doing some free lance research and find that I need to refresh my knowledge of vector calculus a bit. I am having some difficulty with finding web-based sources for the...
Hi, on this page: https://en.wikipedia.org/wiki/Laplace_operator#Two_dimensions
the Laplacian is given for polar coordinates, however this is only for the second order derivative, also described as \delta f . Can someone point me to how to represent the first-order Laplacian operator in polar...
Please help.
I do understand the representation of a vector as: vi∂xi
I also understand the representation of a vector as: vidxi
So far, so good.
I do understand that when the basis transforms covariantly, the coordinates transform contravariantly, and v.v., etc.
Then, I study this thing...
Homework Statement
the line goes through (0, 3/2) and is orthogonal to a tangent line to the part of parabola y = x^2, x > 0
Homework Equations
The Attempt at a Solution
I have problems regarding finding the equation of tangent line to the part of parabola
because the question not...
Homework Statement
How does the gradient of the graph compare to the weight force?
The graph is a Mass vs 1/Acceleration graph (y axis = mass, x axis = Acceleration, It was mentioned to do this.)
Homework Equations
Explain by referring to the formula for Newton's Second Law.
The Attempt at a...
Homework Statement
The problem statement is in the attachment
Homework Equations
E[/B] = -∇φ
∇ = (∂φ/∂r)er
The Attempt at a Solution
I am confused about how to do the derivative apparently because the way I do it gives
E = - (∂[p*r/4πε0r3]/∂r)er = 3*(p*r)/4πε0r4er
I am looking at an explanation of the gradient operator acting on a scalar function ## \phi ##. This is what is written:
In the steps 1.112 and 1.113 it is written that ## \frac {\partial x'_k} {\partial x'_i} ## is equivalent to the Kronecker delta. It makes sense to me that if i=k, then...
Homework Statement
I am self studying relativity. In Wikipedia under the four-gradient section, the contravariant four-vector looks wrong from an Einstein summation notation point of view.
https://en.wikipedia.org/wiki/Four-vector
Homework Equations
It states:
E0∂0-E1∂1-E2∂2-E3∂3 = Eα∂α...
Homework Statement
There is a collection of different force fields, for example:
$$F_{x}=ln z$$
$$F_{y}=-ze^{-y}$$
$$F_{z}=e^{-y}+\frac{x}{z}$$
We are supposed to indicate whether they are conservative and find the potential energy function.
Homework Equations
See Above
The Attempt at a...
Hello,
My professor just gave us a True or False problem that states:
∇H(x,y), the gradient vector of H(x,y), gives us the largest possible rate of change of H at (x,y).
Now, he said the answer is true, but it was my understanding that the gradient itself gives the direction of where the...
Homework Statement
Fc = mv^2/r represents the motion of a simple pendulum. Describe how this data could be graphed so that the gradient of a straight line could be used to determine the velocity of the object.
Homework Equations
Fc = mv^2/r
The Attempt at a Solution
I'm kinda stumped. I tried...
Hello Forum,
Does anybody have suggestions as to how we can use IMU's (accelerometers and gyros) to determine the gradient of a road during a braking event. We have wheel speed inputs so can calculate decelerations independently from the IMU.
Thank You
Tim
My question is mostly about notation. I know the general definitions for divergence and curl, which can be derived from the divergence and Stokes' theorems respectively, are:
\mathrm{div } \vec{E} \bigg| _P = \lim_{\Delta V \to 0} \frac{1}{\Delta V} \iint_{S} \vec{E} \cdot \mathrm{d} \vec{S}...
Can someone please help me prove this product rule? I'm not accustomed to seeing the del operator used on a dot product. My understanding tells me that a dot product produces a scalar and I'm tempted to evaluate the left hand side as scalar 0 but the rule says it yields a vector. I'm very confused
In a river, water flows faster in the middle and slower near the banks of the river and hence, if I placed a twig, it would rotate and hence, the vector field has non-zero Curl.
Curl{v}=∇×v
But I am finding it difficult to interpret the above expression geometrically. In scalar fields, the...
What is the geometrical meaning of ##\nabla\times\nabla T=0##?
The gradient of T(x,y,z) gives the direction of maximum increase of T.
The Curl gives information about how much T curls around a given point.
So the equation says "gradient of T at a point P does not Curl around P.
To know about...
I've come across two different approaches to quantifying what l is in the equation for hydraulic gradient Δh/L. In this first picture L is the parallel distance along the datum across the reference plane
But in this second picture L is the length along the pipe
Why are the two L's...
I learned gradient in 3D space. And gradients where always vectors, pointing in the direction of steepest ... and normal to the surface where the functions is constant.
But reading one-forms , a gradient of a function is not always a vector and it has something to do with metric... Can you proof...
Okay I'm having a little trouble understanding a section of this proof about the product of the gradients of perpendicular lines given in my textbook. I'm gonna type the proof out but there will be a link at the bottom to an online version of the textbook so you can see the accompanying diagram...
Hi,
I am trying to calculate the laplacian of a scalar field but I might actually need something else. So basically I am applying reaction diffusion on a 2d image. I am reading the neighbours, multiplying them with these weights and then add them.
This works great. I don't know if what I am...
1.
Given a function f(x,y) at (x0,y0). Find the two angles the directional derivative makes with the x-axis, where the directional derivative is 1. The angles lie in (-pi,pi].
2.
f(x,y) = sec(pi/14)*sqrt(x^2 + y^2)
p0 = (6,6)
3.
I use the relation D_u = grad(f) * u, where u is the...
Recently I started with multivariable calculus; where I have seen concepts like multivariable function, partial derivative, and so on. A week ago we saw the following concept: directional derivative. Ok, I know the math behind this as well as the way to compute the directional derivative through...