# Tagged Questions

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### Check if two vector equations of parametric surfaces are equivalent

Give the vector equation of the plane through these lines: $\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}4\\1\\1\end{pmatrix}+\lambda\cdot\begin{pmatrix}0\\2\\1\end{pmatrix}\,\,\,$ and ...
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### parametric equations multivariate calculus

could anyone help me to solve this problem Given a parametrization of the tangent line to the curve,(x(t),y(t)) at t=a is: ...
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### Plotting parametric form of a gradient

This is driving me batty. I'm trying to figure out how to plot the gradient of a circle function (is that a vector field?) in parametric form. I don't understand what values to plug in to a get a ...
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### How to parameterise the curve $x^2 = 4y, 3x^3 = 8z$?

As per title, I'm unsure how to parameterise the given curve? Are there different methods? I'm unsure about parameterisation in general, I just tend to remember specific formulas.
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### Parametric Equation of sine wave helically wrapped around a cylinder

I want a parametric equation of a sine wave at a small ramp angle wrapped around a cylindrical body (3D). The parametric equation below gets me close to what I'm looking for, but not quite since the ...
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### Parametric equation of a curve in $R^3$

How to find the parametric equation of the curve in $R^3$, which is the intersection of the sphere of radius $a>0$ centred at the origin, and the plane $x+y+z=0$? I've tried to start looking for ...
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### Finding the speed of a particle (parametric math)

I have to find the speed (as a function of $t$) of a particle whose position at time $t$ seconds is represtented by $$c(t)=(\sin t+t, \cos t+t)$$ How would I go about finding the maximum speed? ...
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### Find the length of the curve

I need to find the length of the curve $$c(t)=(3e^{t}-3, 4e^{t}+7)$$ for $$0\le t \le 1$$ If I understand correctly, I need to take the derivative of the y part of that coordinate over the ...
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### Find the parametric equation to the curve

Find the parametric equation for the curve. $$x^{2}+y^{2}=10$$ I haven't learned parametric equations fully yet, so I wanted to check with you guys and see if you can confirm if I'm doing this ...
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### Express the parametric equation in form of y=f(x)

I need to express the parametric equation in the form of $y=f(x)$ by eliminating the parameter. I haven't learned how to do this yet, I've attempted to read a few pages though but they didn't help me ...
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### How to find the equation of a line which intersects these lines at 90 degrees?

How to find the equation of a line which intersects these lines at 90 degrees? $p\equiv \dfrac{x}{2}=\dfrac{y+1}{0}=\dfrac{z-2}{1}$ $q\equiv \dfrac{x-1}{1}=\dfrac{y-2}{1}=\dfrac{z+5}{0}$ Since the ...
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### Tangent Planes and Surfaces (Calc 3)

I am wondering if I am on the right track for the following question: Find a for the plane $x+y+z=-1$ so that it is a tangent plane to the surface $z=x^2+ay^2$ I figured since you are given a ...
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### Checking a solution of a PDE

I have the following PDE: $$-yu_x + xu_y = 0 \quad\text{where } u(0, y) = f(y)$$ I derived a solution as follows: \begin{align} -yu_x + xu_y =& 0 \\ \iff& ...
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### Evaluate $\iint_s\text{curl}\textbf F\cdot \textbf {n}dS$

Let $\textbf F=<xy,yz,zx>$ and $S$ be the upper half of the ellipsoid $\displaystyle \frac {x^2}{4}+\frac {y^2}{9}+z^2=1$. Evaluate $\iint_s\text {curl}\textbf F\cdot \textbf {n}dS$ I know the ...
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### Parametric equation for a curve $C_1$ in $\mathbb{R}^3$ such that the angle of its tangent $T$ and the $Y axis$ equals the angle of $T$ and a vector

I know that to find the angle between two curves in $\mathbb{R}^3$ at their intersection I differentiate both curves and evaluate at the intersection to find the slope of each tangent line and then do ...
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### Parametrizing a 3D surface

Find a parametrization of the surface $x^3 + 3xy + z^2 = 2$, $z > 0$, and use it to find the tangent plane at $x = 1$, $y = \dfrac{1}{3}$, $z = 0$. I know how to find the tangent plane once I have ...
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### Find the parametric equations of the line of intersection…

Find the parametric equations of the line of intersection of the planes x - z = 1 and x + 2y + 3z = 1. I'm assuming it's something to do with cross product? Here's what I've set up: x y z 1 ...
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### Evaluating a surface integral of a paraboloid

Calculate the average value of $(1+4z)^{3}$ on the surface of the paraboloid $z=x^{2}+y^{2}$,$x^{2}+y^{2} \leq 1$ I'm not sure on how to start this problem. I have already found the area of the ...
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### Supremum Definition in context of a Rectifiable/Differentiable Curve

I am trying to prove that if I have 2 paramterizations of the same curve $\gamma$ and $\sigma$ (i.e. there is continuous bijective map $\phi$ such that $\sigma = \gamma \circ \phi$) then if the curve ...
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### Finding a vector parametric equation given P and Q equations?

Find a vector parametric equation $r⃗(t)$ for the line through the points $P=(3,5,4)$ and $Q=(1,4,7)$ for each of the given conditions on the parameter $t$. If $r⃗ (0)=(3,5,4)$ and $r⃗ (7)=(1,4,7)$, ...
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### Can you give a general expression for the rate of rising water poured in an object?

Given an 3-d object defined by a set of parametric equations (x, y, z), can you write a formula expressing the rate that a liquid rises as it's poured into this object at a constant flow rate? Assume ...
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### There exists a constant arc length parametrization

I heard that for any curve in the plane that can be given parametrically by $\vec{r}(t)=\langle x(t),y(t)\rangle$ for $a\leq t\leq b$ that there exists a constant arc length parametrization, i.e. ...
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### Line integral, Parametrization

I have this line $A=\{(x,y) \in R^2 : y^2+4x^4-4x^2=0\}$ , $(x>0)$ I parametrized it like that : $b(t) = (t, \sqrt{4t^2- 4t^4})$. And my $F$ is $F(x,y) = (x+y,-x)$. But when I calculate my ...
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### Parametrize $|x|+|y|+|z|=1$

How can we parametrize the surface $|x|+|y|+|z|=1$? Here I mean differentiable parametrize. I think we may need to divide it into 8 pieces and consider them respectively.
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### Extend a vector field of normal vectors beyond the surface

I am not terribly well-versed in differential geometry, so please keep that in mind when answering the question. We are given a surface in ${R}^3$ defined parametrically by $\vec{r}(u,v)$ where ...
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### find area of the region $x=a\cos^3\theta$ $y=a\sin^3\theta$

Find the area of the region enclosed by $x=a\cos^3\theta$ and $y=a\sin^3\theta$ What steps should I take in order to find the area?
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### Parametrically defined Spheres in $R^n$

So I have 2 questions here which are closely linked: How do you parametrically define the circle $(x')^2 + (y')^2 = r^2$ using (x') and (y') as coordinates on the plane ax + by + cz = 0 that are ...
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### Testing the multivariable chain rule

Let $x = t$, $y = t^2$, and $z = xy$. Then $$\frac{\partial y}{\partial t} = 2t$$ and since $t = x$, $$\frac{\partial y}{\partial x} = 2x = 2t.$$ Everything is consistent. But now consider using ...
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### Parametric representation of $\sqrt{x^2+y^2}\le z \le 2$

just wondering how to parametrize this. Question is: Let $C$ denote the conical region $\sqrt{x^2+y^2}\le z \le 2$. Find a parametric representation $\mathbf{x}(u,v)$ for $S$, the surface of $C$. ...
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### Parametric Equation along a line segment

I am having some trouble understanding how to determine the parametric equation of a line segment between A(1,1) and B(-1,1). I did some research and came across the following relation: ...
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### Finding surface area of a cone

I will describe the problem then show what I tried to solve it. I need to find the area of the cone defined as follows: $$z^2=a^2(x^2+y^2)$$ $$0\leq z\leq bx+c$$ where $a,b,c>0$ and $b<a$. ...
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Given the following vector equation, how do I eliminate the parameters $u,v$ to get an equation of a surface in rectangular coordinates? $$\vec{r}(u,v)=3u\cos(v)\hat{\imath} + 4u\sin(v)\hat{\jmath} ... 1answer 546 views ### Representing A Plane Curve By A Vector Valued Function I am given the function x^2+y^2=25, and I am suppose to write this as a vector valued function. I have always been awful at these sort of problems, even with parametric equations, which requires ... 0answers 657 views ### Parametrization of square to calculate Dot-product in line-integrals and area-integrals, electric field from \frac{dB}{dt}? I am calculating 3A of Tfy-0.1064 in Aalto University. I realized here that I am misunderstanding something in vector calculus: the thing market in green particularly. I know$$\nabla\times E= ...
$s(t)=(\frac{2}{t^2+1},\frac{2t}{t^2+1})$ I need to calculate a line integral along this path. But I have trouble understanding what it is. I did some googling and it looks that it is a parabola, but ...
I usually use the following parametric equation to find the surface area of a regular cone $z=\sqrt{x^2+y^2}$: $$x=r\cos\theta$$ $$y=r\sin\theta$$ $$z=r$$ And make $0\leq r \leq 2\pi$, \$0 \leq \theta ...