The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with flat polygonal faces), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces. This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration.
A general definition of surface area was sought by Henri Lebesgue and Hermann Minkowski at the turn of the twentieth century. Their work led to the development of geometric measure theory, which studies various notions of surface area for irregular objects of any dimension. An important example is the Minkowski content of a surface.
Problem statement : I draw the problem statement above. I hope I am correct in inferring that the bowl is hemispherical.
Attempt : I could not attempt to the solve the problem. We are given that the rate of change (decrease) in volume is proportional to the surface area ...
Below is an image to calculate the surface area of a sphere using dA. I can see how ##rcos\theta d\phi## works, but I don't understand how that side can't just be ##rd\phi## with a slanted circle representing the arc length. The second part I don't understand is why it is integrated from...
Hello
In a pharmaceutical lab,a certain pressure is applied to IV bags using the equipment shown below.We need to calculate the force acting on the bags based on the applied pressure.I know the formula is F=PxA.But i am not sure what the surface area is.Should i take the whole surface area of...
A water drop of radius ##10^{-2}## m is broken into 1000 equal droplets. Calculate the gain in surface energy. Surface Tension of water is ##0.075 ~N/m##.
So, for the solution of the above problem we need to know how much surface area (combining all 1000 droplets) have increased from the...
My textbook says "A is the area of the circle enclosed by the current" (produced by an electron in a hydrogen atom), A = ##\pi r^2 \sin(\theta)^2##. I don't understand where the ##\sin(\theta)^2## comes from.
Imagine a bubble vibrating in air. Because it vibrates, it's interfacial area increases, thus new molecules are added and removed from the surface as it vibrates.
Consider a molecule is initially at position X_0 at the interface, and over a certain amount of time molecules squeeze and disappear...
I am checking the divergence theorem for the vector field:
$$v = 9y\hat{i} + 9xy\hat{j} -6z\hat{k}$$
The region is inside the cylinder ##x^2 + y^2 = 4## and between ##z = 0## and ##z = x^2 + y^2##
This is my set up for the integral of the derivative (##\nabla \cdot v##) over the region...
Homework Statement
Homework Equations
The Attempt at a Solution
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The solution to this problem is known. I want to use this exercise as a model to understand how to proceed when calculating the surface area of a geometric figure.
Question:
1) Why do we differentiate with...
Homework Statement
Show that the surface and volume element on a deformed sphere are
\sigma = \frac{\rho^2 \sin \theta}{\cos \gamma} d\phi d\theta,
dV = \rho^3 \sin \theta d\phi d\theta,
if \gamma is the angle between normal vector and radius vector.
Homework Equations
n\cdot r = \cos...
I run a vacuum distillation unit that is used to distill ethylene glycol. The old glycol is subjected to 25Hg of vacuum at a temp of 275 degrees F. There are burner tubes submerged in the glycol to heat it up. The vessel has a diameter of 36 inches and is 124 inches long. We fill the vessel...
Here's a crude model rocket fin made out of a material called Coroplast (corrugated plastic):
And here's the exact same fin with the flutes cut out from all around the edges, allowing the edges to be squeezed shut and sealed with tape:
It may look like there are small openings, but...
Knowing that Gauss' law states that the closed integral of e * dA = q(enclosed)/e naut, how would you find exactly what A is in any given problem?
I know it varies from situation to situation depending on the geometry of the charge. For instance, I know that for an infinite wire/line of...
we interest one V-Sorb 2800 BET surface area analyzer, using physical adsorption principle to test particles surface area data, if anyone knows this analyzer principle?
Homework Statement
How large a surface area in units of square feet will 1 gallon of paint cover if we apply a coat of paint that is 0.1cm thick?
Homework Equations
Since Volume is L * W * H and we can assume the object is square besides the height which in this case will be the thickness. So...
I'm given that:
S is the surface z =√(x² + y²) and (x − 2)² + 4y² ≤ 1
I tried parametrizing it using polar coordinates setting
x = 2 + rcos(θ)
y = 2rsin(θ)
0≤θ≤2π, 0≤r≤1
But I'm not getting the ellipse that the original equation for the domain describes
So far I've tried dividing everything...
1. Surfactant molecule is made from water-loving head and grease-loving tail (Figure 1). My question: How do we measure the cross-sectional area of the alkyl chain of surfactant? Do we measure it vertically (refer to GREEN DOUBLE ARROWS of Figure 1) or horizontally (refer to RED DOUBLE ARROWS of...
Just humour me, if you had an infinitely thin cone, would the surface area inside the cone be the same surface area on the outside of the cone? It must be right?
Is there a formula for the surface areas of the inside and outside of a cone WITH thickness?
Hi, I have an mathematics assignment to do, and I wonder if the topic I have chosen is doable for me. I want to minimize the surface area of a cobbler cocktail shaker, and until now my plan was to get the curve equation for the side of it, and get the area equation from surface of revolution...
Homework Statement
Muscle can be torn apart by a force of 100,000 N applied across an area of 1 m2. A 10 cm2 muscle therefore will be torn by a force of 100 N.
If a student of average size were being lowered into a black hole of 1 solar mass, at about what distance from the hole's center will...
I'm building an frame out of solid 5/8 inch thick aluminum square bars. this frame will be around a fire temps close to melting points but there will be insulation in make sure it won't melt.
I was thinking about drilling holes in the free space of the aluminum to reduce it's weight and if you...
For the following three-dimensional surface, z = -4.53 + 2.67x + 2.78y - 1.09xy, I would like to calculate the area for each of three subsections of this surface: (1) for which z is in between the corresponding x and y values (i.e., x < z < y OR y < z < x); (2) for which x is in between the...
Hey everyone,
I was recently reading a paper on surface enhanced Raman scattering, and it mentioned that plasmons (and for that matter surface plasmon polaritons-where my interest lies) are sensitive to the surface to volume ratio of the structure. I can begin to understand intuitively with...
I'm trying to follow Schwabl Thermodynamics, and I found the following equation for the surface area of a unit d-sphere:
$$ \int d\Omega_d = \frac{2 \pi^{d/2}}{\Gamma(d/2)} $$
But this formula clearly fails for d=1:
should be $$\pi$$
and d=2:
should be $$ 4 \pi $$. What gives?