Line Definition and 1000 Threads

According to the formula, an increase in numerical aperture leads to a decrease in minimum line width and thus better resolution. However, if were to draw it out, given the same depth of focus, why does the minimum line width increase with higher numerical aperture?

When I take ##x = 2\cos(t)## and ##y = 2\sin(t)##, the integral becomes ##\int_{t=\frac{\pi}{2}}^0 4(2\cos(t))^2 \cdot 2 dt = -8\pi##. The final answer is ##8\pi##. Why is my method wrong? I played around with desmos and the parameterisation seems correct...

How did you find PF?: via Google search Hi, I wonder if the shade from the sun on the ground, on a sunny day, is it a straight line or a curved line?? and if it is curved, how curvature can be calculated ?? Let´s say I have a pole or building, and I can follow up the shade of its top on the...

My approach - i think similar to ms approach. The required Equation will be in the form ##y=mx## ##\begin{pmatrix} a & b^2 \\ c^2 & a \end{pmatrix} ⋅ \begin{pmatrix} k \\ mk \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} ## ##ak+b^2mk=x## ##kc^2+amk=y## ##x=k(a+b^2m)##...

I am trying to run multiple lines input in Mathematica in the free-form input. From everything I read online, you simply hit "enter", but hitting enter runs my process. Both enters, the one next to my letters, and the one on the numerical keypad. Hitting shift+enter also runs it. I can't...

For ##c > 0## and ##0 \le x \le 1##, find the complex integral $$\int_{c - \infty i}^{c + \infty i} \frac{x^s}{s}\, ds$$

X 0 50 100 150 200 250 Y 120 178 201 249 303 364 XY 0 8900 20100 37350 60600 91000 X^2 0 2500 10000 22500 40000 62500 ∑X 750 ∑Y 1415 ∑XY 217950 ∑X^2 137500 M = (6*217950)-(750*1415) / (6*137500) - (750)^2 M = 1643 / 1750 M =...

I stumbled upon this video on YouTube: Here is a screenshot with some colored lines added of the part that generated a few questions in my head that I hope some of you smart folk can answer for me. In the video, the spherical magnet (at the end near the two big magnets) appears to increase...

I have drawn the I-V characteristic graph of the diode. I am facing problems with drawing the load line. For what value of E (Source Voltage), should I draw the Load Line? To get the I-V graph, I had to continuously change E.

I was thinking of using the sphere and point charge as an analog, but is quite diferent from what i have seen

For this problem, The solution is, However, I could not get that. By getting the system in REF, I got ##x = 3 + \frac{3}{7}z ## and ##y = \frac{1}{7}z##. Therefore z is a free variable so ##x = 3 + \frac{3}{7}t = 0## and ##y = \frac{1}{7}t##. Thus equation of line is ##x\hat i + y\hat j +...

Hi all, I am a beginner in Linear Algebra. I am solving problems on vector spaces and subspaces from the book Introduction to Linear Algebra by Gilbert Strang. I have come across the following question: Suppose P is a plane through (0,0,0) and L is a line through (0,0,0). The smallest vector...

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...

For this problem, The limiting position of R is (4,0). However, I am trying to solve this problem using a method that is different to the solutions. So far I have got, ##C_1: (x - 1)^2 + y^2 = 1## ##C_2: x^2 + y^2 = r^2## To find the equation of PQ, ## P(0,r) ## and ##R(R,0) ## ## y =...

Probability of any random n points on a line being within a given distance Hi, I am a software engineer trying to solve the following problem analytically given a line segment in cm and n random points on it what is the probability that the distance between any 2 consecutive points on the...

According to Maxwell’s Equations, the speed an EM plane wave in free space, far from its source, is determined by the electric constant, ε0, and the magnetic constant, μ0, such that c = 1/√( ε0 μ0). The units of ε0 are capacitance per unit length and the units of μ0 are inductance per unit...

Water line on a certain property has been busted for 2 years and no water in line. I am calling plumber tomorrow to get fixed and am just curious before I do that if fact that there has been no water in line would pose any particular issue to plumber just coming out and fixing that one spot...

Hello! In the following image, is it true that S1 = mg/2. Thanks for answears!

Is there any way to construct a line segment, that has the lenght of the circumference of a circle using only a ruler and a compass? My intuition says "no" Or phrasing the question in another way: given two line segments, can I prove, that the longer line segment has the length of the...

I'm assuming the boat is traveling north at an 8 m/s and the river is flowing east at a 2 m/s. For the boat to move in a straight line, it has to aim at an upstream angle given by #theta#. Using SOH CAH TOA, ##v_r = 2m/s##, ##v_b = 8m/s## $$\theta =\sin^{-1} \frac{v_r}{v_b}$$ $$\theta = 14...

Can we say, (y + z ) x1 = (y1 + z1) x is also an equation of a straight line in 3 dimensional space, where (x1,y1,z1) and (x,y,z) are the coordinates of a given point and a variable point respectively on a 3D line that passes through the origin, have seen equation of a straight line in 3...

Okay so I am a little confused as to where I made a mistake. I couldn't figure out how to program Latex into this website but I attached a file with the work I did and an explanation of my thought process along the way.

This is a tricky and difficult question for me. I know from reading various textbooks that electric lines of force are always continuous without breaks, but cannot pinpoint a reason for this. The only reason I can come up is that an electric line of force must always begin and end on charges...

TL;DR Summary: Find the electric field of a long line charge at a radial distance where the potential is 24V higher than at a radial distance r_1=3m where E=4V/m. Answer: 29.5V/m. Never mind: I retract this question. The integral apparently is supposed to diverge! I apologize for not reading...

First of all i tried to follow the textbook. Here they start of by modelling the atom as an harmonic oscilator: Then they find the solution as: They neglect the second term as omega_0 >> gamma which also makes good sense so they end up with: So far so good. After this they state the...

so this is what the FBD is.... but to be fair, to me this one looks as if the normal force in the direction of the radial line, yet it isn't???? here in the solution, it's not along the radial line, whys that???

so I was wondering. there is this normal force on the can from the path. And there's this formula to find the angle between the radial line and the tangent or also between the normal force and either the radial or theta axis. the formula is ##\psi = r/dr/d\theta##. The thing is that here they...

I - A point divides a line into two parts; II - A line divides a plane into two parts; III - Does a smaller sphere divide a larger sphere into two parts, like layers of an onion? Note that the first two statements, the question of infinity must be considered. For the third statement, is the...

Q：Please hlep me to understand which ans is correct.To determine the flow rate in Line AB. $$\mathrm{Known}：V_A,q,r_A = constant.$$ so/ select：## A,{B}^{\text{'}},B,A,## is control volume $${Q}_{AB}={Q}_{A{B}^{\text{'}}}=\iint _{A}^{}({V}_{A})dA={\int }_{{\theta }_{A}}^{{\theta...

I would like to read a specific line of a file. Everything I have seen does it by reading the file from the top, ignoring what is read until the line of interest is reached. Is it possible to move the pointer of the reader to the line directly, say by counting the number of line end sequences...

In the old days, people plugged landline phones into house/apartment wall mounted phone jacks. This was standard. Then, at some point, people started plugging their landline phone cords (via an adapter) into their internet modems. Dumb question for a non-techie. Why is that? Also: a.)...

##curl([x^2z, 3x , -y^3],[x,y,z]) =[-3y^2 ,x^2,3]## The unit normal vector to the surface ##z(x,y)=x^2+y^2## is ##n= \frac{-2xi -2yj +k}{\sqrt{1+4x^2 +4y^2}}## ##[-3y^2,x^2,3]\cdot n= \frac{-6x^2y +6xy^2}{\sqrt{1+4x^2 + 4y^2}}## Since ##\Sigma## can be parametrized as ##r(x,y) = xi + yj +(x^2...

I just found out about Saudi Arabia's "The Line", and I don't want to mock it with my very limited knowledge (I honestly would have thought it was satire), or bring up politics behind it, or go into its lofty goals like 100% clean energy or life enhanced by AI. To me, the first question is just...

Here is a picture of the problem: I honestly am pretty lost, I'm not looking for an answer, more so an idea to get me started. But here is what I was thinking: In the equation above I was trying to use: For U I am unsure how to incorporate the weight of the blocks into the u, so I am unsure...

From Wikipedia article about Hyperbolic motion, I have the following coordinate equations of motion for Bob in his accelerated frame: ##t(T)=\frac{c}{g} \cdot \ln{(\sqrt{1+(\frac{g \cdot T}{c})^2}+\frac{g \cdot T}{c})} \quad (1)## ##x(T)=\frac{c^2}{g} \cdot (\sqrt{1+(\frac{g \cdot T}{c})^2}-1)...

Let ##f : \mathbb{R} \to \mathbb{R}## be a uniformly continuous function. Show that, for some positive constants ##A## and ##B##, we have ##|f(x)| \le A + B|x|## for all ##x\in \mathbb{R}##.

Author's answer: Recognizing that this integral is simply a vector line integral of the vector field ##F=(x^2−y^2)i+(x^2+y^2)j## over the closed, simple curve c given by the edge of the unit square, one sees that ##(x^2−y^2)dx+(x^2+y^2)dy=F\cdot ds## is just a differentiable 1-form. The...

There is a power line of undetermined voltage hanging just 4 feet above my RV and the stream running next to us. If during the rain, a branch from the redwood tree above us falls and causes the hot line to land on the RV and stay there, is there a way to prevent us being shocked while we try to...

My answer:

t=0 => v(0) = 4(0) - 3(0)^2 = 0m/s t=2 => v(2) = 4(2) - 3(2)^2 = -4m/s Vavg => (v(0) + v(2))/2 = -2m/s When researching the answer, I noticed that they used integrals to solve this question. The only problem is that we never learned about integrals/ derivatives or anti derivative. Is there any...

IMPORTANT: NO CALCULATORS I assumed two points, (a, f(a)) and (b, f(b)) where b is greater than a. Since the tangent line is shared, I did f'(a) = f'(b): 1) 4a^3 - 4a - 1 = 4b^3 - 4b - 1 2) 4a^3 - 4a = 4b^3 - 4b 3) 4(a^3 - a) = 4(b^3 - b) 4) a^3 - a = b^3 - b 5) a^3 - b^3 = a - b 6) (a...

Consider the line element $$dl^2=d\theta^2 + \sin^2\theta\, d\varphi^2$$ where ##\theta\in [0,\pi]##. The standard interpretation of this line element is to take ##\varphi\in [0,2\pi)##, in which case the line element represents the standard metric of the sphere ##S^2##. However, from the line...

I know the Gauss law for surface integral to calculate total charge by integrating the normal components of electric field around whole surface . but in above expression charge is calculated using line integration of normal components of electric field along line. i don't understand this...

A 20 minute video exploring Population Density, Topography, Orography, and Rainfall. The narrator is a bit annoying but there is a lot of information in it.

How do we define tangent line to curve accurately ? I cannot say it is a straight line who intersect the curve in one point because if we draw y = x^2 & make any vertical line, it will intersect the curve and still not the tangent we know. Moreover, tangent line may intersect the curve at other...

The question is as shown below. ( Text book question). The textbook solution is indicated below. Discussion; Now they seemingly used ##r=1## to arrive at ##x=0.8+0.2y##. That is, ##y=-4+5x## then, since ##r=1##, ...implying perfect correlation therefore, ##5x=4+y## ##x=0.8+0.2y## My other...

∫zds=∫acos(t)*( (acos(2t))^2+(2asin(t))^2+(-asin(t))^2 )^1/2 dt , (0≤t≤pi/2) Simplified : ∫a^2cos(t)*(cos^2(2t)+5sin^2(t) )^1/2 dt , (0≤t≤pi/2) However here i get stuck and i can´t find a way to rewrite it better or to integrate as it is. Can i please get some help in this?

I have solved the gradient: gradf(2,-1)=(4,2) and have the tangent plane: 4x+2y+3=0 Somehow the answer is: 3=2x+y And i really don´t understand why.

So, I am able to calculate the electric potential in another way but I know that this way is supposed to work as well, but I don't get the correct result. I calculated the electric field at P in the previous exercise and its absolute value is $$ E = \frac {k Q} {D^2-0.25*l^2} $$ This is...