# Tagged Questions

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### Spivak's “Calculus in Manifolds” problems

I have some troubles with this problems. Problem 1.18: If $A \subset [0,1]$ is the union of open intervals $(a_i,b_i)$ such that every rational number of $(0,1)$ is contained in $(a_i,b_i)$, for ...
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### Line integral over a curve in the II quadrant

I am lost here: $C = x^2 + y^2 = 4$ from $(0,2)$ to $(-2, 0)$. Calculate $\ \int_c y^2 ds \ \$ and give reasons the sign is correct. It's obviously the circular arc going counterclockwise from ...
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### Question about vector equations of lines and planes

Find the equation of the line going through the point $(2,-3,4)$ ,and which is perpendicular to the plane $x+2y + 2z = 13$ So I tried this: the normal of the plane is $(1,2,2)$, random point on the ...
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### Vectors Question

I have a question regarding Vectors; Find the equation of the plane perpendicular to the vector $\vec{n}\space=(2,3,6)$ and which goes through the point $A(1,5,3)$. (A cartesian and parametric ...
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### normal of a function

Function $$\mathbf{r}(u,v)=(2u+2v)\mathbf{i}+(-3+v^2)\mathbf{j}+(2u^2)\mathbf{k}$$ is a parametrization of a surface. What's the normal vector of this surface at the point $(u,v)=(1,-2)$. What I got: ...
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### Field lines of vector field

Okey if $\phi(x,y)=\ln(x^2+y^2), (x,y) \neq (0,0)$. Find the field lines for $\mathbf{G}=\nabla \phi$. So $\mathbf{G}=\frac{2x}{x^2+y^2}\mathbf{i}+\frac{2y}{x^2+y^2}\mathbf{j}$ right? To find the ...
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Let $C$ be curve along surfaces $z=\ln(1+x)$ and $y=x$ from $(0,0,0)$ to $(1,1,\ln(2))$. Calculate the work done by vector field $$\mathbf{F}(x,y,z)=(2x\sin(\pi y)-e^z)\mathbf{i}+(\pi x^2 \cos (\pi ... 2answers 27 views ### Question on vector fields Which ones are vector fields? (I checked my answers) Temperature of room at given point The gravitation that object with mass creates (x) The density of an object at given point Function f: ... 1answer 36 views ### Fractional change in volume from scale-factor I was given the following question which I am unable to get a seemingly correct answer from: A body expands linearly by a factor \alpha due to an increase in temperature. Because of the ... 1answer 34 views ### A particle has the following path: \vec{r}(t)=t^2\hat{i}+(t^3-4t)\hat{j} A particle has the following path:$$\vec{r}(t)=t^2\hat{i}+(t^3-4t)\hat{j}At $t_0=2$ the particle fudges (leaves by the tangent). What is the position of the particle at $t=3$? The particle has ...
Find the potential function of a conservative vector field $\mathbf{F}(r,\phi)=r\sin(2\phi)\mathbf{\hat{r}}+r\cos(2\phi)\mathbf{\hat{\phi}}$ Does my solution seem right: So if ...
I am having a hard time finishing this problem up: Consider the surface $4 x^{2} + 9 y^{2} + 4 z^{2} = 17$ and the point $P = \left( 1, 1, 1 \right)$ on this surface. A) Find the outward unit ...