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...

8. ### I What is the gradient of a divergence and is it always zero?

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...
9. S

### I What is the gradient in polar coordinates?

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...
10. ### A How do I see a dual basis

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...
11. ### Find the equation of a tangent line to y = x^2?

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...
12. ### How does the gradient of the graph compare to the W force

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...
13. ### Finding the Electric Field given the potential in spherical

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
14. ### I Kronecker Delta and Gradient Operator

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...
15. ### Contravariant Four-gradient ESN in Wikipedia appears wrong

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...
16. ### Figuring Out if A Force Field is Conservative or Not

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...
17. ### I Gradient Vector- largest possible rate of change?

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...
18. ### How to graph Fc=mv^2/r so that the gradient = velocity

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
20. ### Div and curl in other coordinate systems

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}...
21. ### Proof of product rule for gradients

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
22. ### Definition of curl

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...
23. ### Geometrical meaning of Curl(Gradient(T))=0

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...
24. ### I Confusion Over Hydraulic Gradient, L parameter

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...
25. ### Why is a gradient not always a vector

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...
26. ### Derive grad T in spherical coordinates

Homework Statement ##x=r\sin\theta\cos\phi,\,\,\,\,\,y=r\sin\theta\sin\phi,\,\,\,\,\,z=r\cos\theta## ##\hat{x}=\sin\theta\cos\phi\,\hat{r}+\cos\theta\cos\phi\,\hat{\theta}-\sin\phi\,\hat{\phi}## ##\hat{y}=\sin\theta\sin\phi\,\hat{r}+\cos\theta\sin\phi\,\hat{\theta}+\cos\phi\,\hat{\phi}##...
27. ### Product of the gradients of perpendicular lines proof help

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...
28. ### How to get the laplacian of a scalar field?

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...
29. ### Find two angles where the directional derivative is 1 at p0

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...
30. ### Directional derivative and gradient definition confusion

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...