surface

1. How do I find the work done by the kinetic friction force when the surface is a curve

The problem is as follows: The diagram from below shows a boy of 30 kg of mass slides down a slide from a height of $5\,m$ starting from rest in point $A$. He reaches to point $B$ with a speed of $4\,\frac{m}{s}$. Find the work done by the frictional force in Joules. You may use...
2. Name of surface?

What is the name of a surface in hyperbolic space whose points are a fixed distance from a given plane in that space? Where can I find more information for such a surface?
3. Line, Surface, and Volume elements- Parameters

Line, Surface, and Volume elements. Let r be a vector in space r=(x,y,z) LINE x,y,z given in terms of parameter u. dr=r'du. dL=|r'|du SURFACE x,y,z given in terms of parameters u,v. In any coordinate direction, holding the other fixed, dr=r_{u}du, dr=r_{v}dv. dA=|r_{u}Xr_{v}|dudv...
4. Calculate the area of a quadrilateral

While thinking of strategies ... ;) I manage to solve the special case when M is on the half of the arc BC. The answer is: P_{APQD}=\frac{a^{2}}{2}. My reason was that because it does not matter where on arc BC is M, the area must be the same wherever it is positioned. I'm concerned that this...
5. Area of a Surface

I have solved this problem. I wish to find out if my solution is correct. **Problem:** Determine the area of the surface $A$ of that portion of the paraboloid: $$x^2+y^2-2z=0$$ where $x^2+y^2\le 8$, $y\ge x$ **Solution:** From the surface: $x^2+y^2-2z=0$...
6. Calculus 3 -Quadric Surface Question

Hi! I'm new here. I wanted help on a simple "Quadric Surface" Question from calculus 3. Our teacher in college didn't explained it in the class. Convert it into the standard form and sketch it: $x^2+y^2+z^2=3z$ Thanks!
7. Sine and Cosine to find area of surface

I am struggling with this one, finding the area of the land via the sine or cosine rule. I just don't see how I can go about it, I know I could use the formula for the area of a non-right angled triangle if I split it from B to D and make two triangles (Â½ ab sin C) However, I don't see how I...
8. surface integral problem

Find the area of the cylinder x^2 + z^2 = a^2 that lies inside the cylinder x^2 + y^2 = a^2. Official Answer: 8a^2 My solution x = r cos \theta , y = r sin \theta , z = \sqrt{a^2 - r^2cos^2 \theta } \int_{0}^{2 \pi} \int_{0}^{a} \sqrt { \frac {(r^2cos \theta sin^2 \theta + r^2 cos^3...
9. Proof of Surface Area and Volume of a sphere Using Integral Calculus

Hi! Letâ€™s consider a sphere with a radius r. What's his volume and his area. The full answer is given in the following link : https://smartmanmaths.com/2017/09/22/proof-of-surface-area-and-volume-of-a-sphere-using-integral-calculus/ Thanks and enjoy your mind!
10. Surface area via integration confusion

Hello forum, I am struggling on this problem. I am asked to evaluate $\oint \vec r \dot \,d\vec \sigma$ over the whole surface of the cylinder bounded by $x^2+y^2=1, z=0, z=3$ It seems pretty straight forward geometrically as it is just a unit circle at $z=0$ and then it extends...
11. 3-D Surface plot of equation

What does the plot of this surface equation look like ? I know it has a shape of the sphere but I am confused what is 1 and 4 and what they represent.
12. Curve vs Surface

I have a definition for curve and surface in my notebook that I don't really understand. Below are screenshots from the definitions and I would be really thankful if somebody explains them to me. Or you can explain to me what curve and surface is the way you know it. Thank you :D
13. parameterization in Surface integration

I want to surface integrate over a surface of the plane S : 2x+3y+6z =12 which lie in the 1st octant. Should I use parameterization, if I should, how?
14. requesting for explanation of surface integration

Evaluate âˆ«âˆ« < x, y, -2 > * N dS, where D is given by z = 1 - x^2 - y^2, x^2 + y^2 <= 1, oriented up. Official Answer: - pi Cross Product: \int_{0}^{2\pi} \int_{0}^{1} ( r cos \theta , r sin \theta , -2 ) * ( 2r^2 cos \theta , 2r^2 sin \theta , r ) ~ dr d \theta \int_{0}^{2\pi}...
15. Another Surface Integral problem... #2

Evaluate âˆ«âˆ« ( 2, -3, 4 ) * N dS, where D is given by z = x^2 + y^2, -1 <= x <= 1, -1 <= y <= 1, oriented up. Official Answer: 60 My solution: x = r cos \theta y = r sin \theta z = r^2 G(r, \theta) = ( r cos \theta , r sin \theta , r^2 ) dG/dr = ( cos \theta , sin \theta , 2r )...
16. Generic surface integral problem

Find the centroid of the surface of a right circular cone of height h and base radius r, not including the base. I don't even know how to find the two variables gradient of <x,y,z>..
17. Another Surface Integral problem...

Find the center of mass of an object that occupies the surface z = sqrt(x^2+y^2), 1 <= z <= 4 and has density z * x^2. Official Answer: ( 0, 0, 2275/682 ) Here is my solution: x = r cos \theta y = r sin \theta z = \sqrt { r^2 cos^2 \theta + r^2 sin^2 \theta } = r...
18. Flux of the vector field through the surface

Hello! I need help with determining the flux of the vector field. I'm enclosing link to my math.stackexchange question. https://math.stackexchange.com/questions/2382195/determine-the-flux-of-the-vector-field-trough-the-surface
19. Minimize surface area for tank

Hello forum. I am trying to make a rectangular tank that has a volume of 500 cubic feet, but minimizes surface area. The rectangular tank does not have a lid. $V = xyz = 500$ $\Rightarrow z = \frac{500}{xy}$ Surface area can be described as: $S = 2yz + 2xz + xy$ This accounts for the walls...
20. Tangent plane to surface

The questions says to find tangent plane to the surface x^2-y^2-3z=0 that goes through A(0,0-1) and is parallel to line : (0,0,0)+t(2,1,2). I'm really stuck here.