0
votes
2answers
67 views

(I only need some hints)Find the vector equation of a space curve that represents an ellipse with the given center that lies in the given plane

Full disclosure, this is for a Calculus III graded homework set--though we are allowed to use any resources available to complete it. I feel I have a good understanding of space curves, though my ...
0
votes
4answers
115 views

surface that is created by the intersection of paraboloid and plane

Find the surface that is created by the intersection of the paraboloid $x^2+y^2-z=0$ and the plane $z=2$. $$x^2+y^2-z=0 \Rightarrow x^2+y^2=z$$ $$z=2$$ EDIT: I had to find the area of the surface ...
-1
votes
1answer
42 views

Find the equation of a plane [duplicate]

Find the equation of a plane that passes through point $P(1,5,1)$, and is perpendicular to the planes $2x+y-2z=2$ and $x+3z=4$ My only guess so far is that we can obtain the plane's normal vector ...
-1
votes
2answers
589 views

How do you find the acute angle between the lines: $x+2y=7$ and $5x-y=2$. [closed]

Find the acute angle between the lines: $x+2y=7$ and $5x-y=2$. Use vectors.
3
votes
0answers
71 views

Clarification of the Jacobian

Well, that was cool if not tedious but I understand the Jacobian and its application to changing coordinate systems. $${J_{POLAR}= \rho}$$ $$ {J_{cyl}= \rho}$$ and $${J_{sphere}=\rho^2\sin\phi}$$ ...
1
vote
1answer
83 views

Why is the Jacobian ${\rho}$

Just a little confused. When I find the volume of a cone (or a sphere) for that matter I multiply the partial derivatives by the Jacobian. ${\rho}$ for a cone. and ${\rho^2 \sin \phi}$ for a sphere. ...
1
vote
1answer
76 views

Trying to understand Volume of a cone without the unit sphere

I have been working on the double integral proof for the volume of a cone. I found that I can use a unit-sphere Where the base of the cone is the equator and the height is the distance ${\rho}$ to ...
0
votes
1answer
100 views

Can't find the derivation ${\rho^2\sin\phi}$

I have accepted that the equation of a sphere in spherical coordinates is ${\rho^2\sin\phi}$. The triple integral is just to nice. What I don't understand is what happened to ${\theta}$. How can you ...
2
votes
1answer
163 views

Geometric Interpretation of Jacobi identity for cross product

Is there a geometric "reason" for the Jacobi identity for cross products? Some geometric equality of some area ...? All proofs I know work by some form of linear algebra (or use the interpretation as ...
0
votes
1answer
56 views

boundary of the half-annulus

Question Let C be the boundary of the half-annulus a^2 < (x^2 + y^2) < b^2 and y>=0 in the x-y plane, traversed in the negative direction what is ∫ (5e^(-7x^2) - y^3) dx + x^3 +cosh^2(5y) dy ...
1
vote
2answers
221 views

Parametrization of $x^2+y^2=z^2$

How can we show that any point on $x^2+y^2=z^2$ can be written in the form (z $\cos(\theta)$, z $\sin(\theta)$, z) for some $\theta$? Here is how I tried to approach it: $$(z \cos(\theta))^2+(z ...
0
votes
2answers
139 views

How many times can quadric kiss cosine at given point?

Let a quadric $ax^2+2bxy+cy^2+dx+ey+f=0$ touches the plot of $y=\cos(x)$ at the point $(0,1)$ with multiplicity $n$. What is the maximum possible value of $n$? Recall that a joint point $P$ of ...
1
vote
2answers
74 views

Being inside or outside of an ellipse

Let $A$ be a point $A$ not belonging to an ellipse $E$. We say that $A$ lies inside $E$ if every line passing trough $A$ intersects $E$. We say that $A$ lies otside $E$ if some line passing trough $A$ ...
0
votes
2answers
96 views

The closes point to a curve in space.

I am working on the following problem. Find the point closest to the origin, of the curve of intersection of the plane $2y+4z =5$ and the cone $z^2 = 4(x^2+y^2)$ I was able to see that the ...
1
vote
0answers
88 views

Describing domain of integration of triple integral

I'm struggling to visualize the following problem: This question concerns the integral $\int_{0}^{2}\int_0^{\sqrt{4-y^2}}\int_{\sqrt{x^2+y^2}}^{\sqrt{8-x^2-y^2}}\!z\ \mathrm{d}z\ \mathrm{d}x\ ...
2
votes
3answers
2k views

surface area of a sphere above a cylinder

I need to find the surface area of the sphere $x^2+y^2+z^2=4$ above the cone $z = \sqrt{x^2+y^2}$, but I'm not sure how. I know that the surface area of a surface can be calculated with the equation ...
1
vote
0answers
111 views

Parametrizing a section of a torus

Consider the torus obtained by rotating the circle $(x-R)^2+z^2=r^2$ about the $z$-axis, where $R>r>0$. Parametrize the part of this torus where $z>x+y$. My approach to this so far is to ...
3
votes
1answer
161 views

The intuition behind the definition of geodesics on a Riemannian manifold. (A non-technical question)

In the text I'm studying, the idea behind the definition of a geodesic on a Riemannian manifold was sketched via paths in $\mathbb{R}^n$. I have trouble understanding some aspects of it. Let $\gamma: ...
1
vote
1answer
245 views

shortest distance between two points on $S^2$

Length of Curve in $2D$ is $l_{\gamma}(\mathbb{R}^2)=\int_{0}^{1}\sqrt{(dr/dt)^2+r^2(d\theta/dt)^2}$ Length of a curve in $3D$ is ...
4
votes
2answers
400 views

Length of curve in 3D spherical coordinate

let $r$ be the magnitude of a vector in 3D with Spherical co-ordinate $(r,\theta,\phi)$ and cartesian coordinates is $(x,y,z)$, whose angle with $z$ axis is $\phi$ and projection of the vector makes ...
1
vote
2answers
325 views

How can I calculate the Euclidian displacement of two places on a sphere (earth in this case ) and calculate the

I would like to get the formula on how to calculate the distance between two geographical co-ordinates on earth and heading angle relative to True North. Say from New York to New Dehli , I draw a ...
2
votes
0answers
356 views

Proof of coarea formula

I want to prove the coarea formula $\operatorname{Vol}(M) = \int_M d\operatorname{Vol}_M = \int_{-\infty}^\infty \frac{1}{|\nabla f|} \operatorname{area }(f^{-1}(t)) dt$ where $f: M \rightarrow ...
0
votes
1answer
213 views

Difficult volume computation inside an ellipsoid and above a plane

I am taking a Calculus course currently and am stuck on the last question of my assignment. Find the volume of the region inside the ellipsoid $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} ...
1
vote
1answer
142 views

Parametrization of a solid

Find a parametrization $\sigma : I \subseteq \mathbb{R}^3 \rightarrow \mathbb{R}^3$, with $I$ a parallelepiped, of $\lbrace (x,y,z) \in \mathbb{R}^3 : |z| \leq 4x^2 + 9y^2 \leq 1 \rbrace $.
1
vote
0answers
112 views

Circle-Circle intersection coordinate system

Consider two points in the 2D Euclidean plane, the origin $0$ and $x$. One can define a co-ordinate system such that for any point $y$ in the plane, $y$ is parametrized by its distance from $0$, call ...
0
votes
2answers
214 views

Help me with Cylinder -coordinates problem, back to Cartesian or not? How to do it fast?

Source of the problem, 3b here. Problem Question Electricity density in cylinder coordinates is $\bar{J}=e^{-r^2}\bar{e}_z$. Current creates magnetic field of the form ...
3
votes
1answer
128 views

What volume does $2x \le x^2+y^2+z^2 \le 4x$ represent?

I'm evaluating $\iiint_V f(x,y,z) dV$ where V is defined by $$2x \le x^2+y^2+z^2 \le 4x $$ To simplify things I swapped x and z, and moved to spherical coordinates: $$ 0 \le \theta \le 2\pi, 2 \cos ...
0
votes
1answer
760 views

Find point on sphere with directional tangent vector

Say a sphere equation like this: $x^2+y^2+z^2=5$. I want to find a point on the sphere whose tangent vector is perpendicular to the vector $\begin{bmatrix} 2\\ 3\\ 4 \end{bmatrix}$. I go ...
-1
votes
2answers
465 views

How do I add Vectors in Vector Calculus?

For example: Suppose vector u = (-2,3) and vector v = (-5,3) then: (u + v = ?) and (u - v = ?) and (v - u = ?) and (6u = ?) and (-1/8v = ?) and (3u - 4v = ?)
1
vote
1answer
494 views

Locus of osculation of concentric ellipses (elliptic pond ripples)

If you dropped two rocks in a pond, the concentric circles emanating from the two spots would osculate $\infty$ times. The locus of osculating points would form a line. Now imagine that instead of ...