0
votes
1answer
25 views

Let $V$ be the space of real sequences {${x_{1},x_{2},…}$} so that $\sum_{k=1}^\infty {x_{k}}^{2}$ converges. Prove that this space is not numerable

Let $V$ be the space of real sequences {${x_{1},x_{2},...}$} so that $\sum_{k=1}^\infty {x_{k}}^{2}$ converges. Prove that this space is not numerable: My attempt: I have already proved that this is ...
1
vote
0answers
21 views

Acute angle between plane and line

Find the acute angle between: $x-y-3z=5$ and $x=2-t$ $y=2t$ $z=3t-1$ Here is how I proceed. I take the dot-product of the normal of the plane and the directional vector of the line. This gives me ...
0
votes
1answer
43 views

I need help with linear transforms? Linear Algebra [closed]

In the question below, how was [T]ff found? I have tried but I can't understand how because I usually start from a given matrix with variables, but non is given here. website is here; ...
1
vote
0answers
36 views

Faster way of finding critical points?

So I am looking at parametric vector function. $$ \begin{vmatrix} \cos (t) & -\sin (t) & 0 \\ \cos f(t) \sin (t) & \cos f(t) \cos (t) & -\sin f(t) \\ ...
0
votes
2answers
122 views

curl of (cross product of two vectors), i know the formula, but not sure how to prove it

$$\text{curl } \left(\textbf{F}\times \textbf{G}\right) = \textbf{F}\text{ div}\textbf{ G}- \textbf{G}\text{ div}\textbf{ F}+ \left(\textbf{G}\cdot \nabla \right)\textbf{F}- \left(\textbf{F}\cdot ...
1
vote
1answer
30 views

Proof Green's theorem $F(x,y)=(x-y)i+xj$

I was reading on Green's theorem and have appreciated the concept. Given a question, I think, I can solve it.But I came across a question that reads: Verify the Green's theorem for the vector given ...
0
votes
1answer
21 views

Verifying The Divergence Theorem

Q: "Given the cylindrical region $x^2 + y^2\leq 1 $ ,where $ 0\leq z \leq 1 $, and the vector field $\underline{F} = 3x\underline{i} -5z\underline{k} $, verify the divergence theorem." For the ...
0
votes
1answer
27 views

evaluating a flux integral

Question: "Region V, of unit volume, is bounded by the closed surface S. Given the vector field $\mathbf{F}=\langle 7x,2y,5z\rangle$, evaluate: $$\int_S \mathbf{F}\cdot\mathbf{dS}$$ I guessed that ...
1
vote
1answer
56 views

Using Cylindrical Coordinates to Compute Curl

I am given a vector field $\vec{A} = A_\rho \space \hat{e_\rho} + A_\phi \space \hat{e_\phi} + A_z \space \hat{e_z}$, and I am supposed to use the unit vectors (provided below) in cylindrical ...
0
votes
1answer
22 views

multivariable calculus question norm

Given vector space C([a,b],$ \mathbb{R} $) of continuous functions of [a,b] in $ \mathbb{R}. $ Prove that the function $ \left \| f \right \|_{1}=\int_{a}^{b}\left | f(t) \right |dt $ is a norm. Also ...
0
votes
2answers
56 views

Multivariable calculus - find total derivative

I want to find the total derivative of the function $f: \mathbb R^n \to \mathbb R^n$, $f(x)=\frac{x}{|x|}$ If I was to copy what the teacher taught, I should find the limit of $\lim_{t \to 0} ...
1
vote
0answers
41 views

Getting “semi” orthogonal basis from a linear independent set

Let $K_i: \mathbb{R}\mapsto \mathbb{R}^k$ are continuous functions for all $i=1,\dots,k-d$ such that for every fixed $t\in\mathbb{R}$ we have ${\cal K}_t=\{K_1(t),\dots,K_{k-d}(t)\}$ be a linear ...
1
vote
1answer
26 views

$g''(t_0)$ where $g(t) = f(t, 1-t) , t_0 = 0$

Let z = f(x,y) be a differentiable function such that $\frac{∂f}{∂x}$ (0,1) = -1 ,$\frac{∂f}{∂y}$ (0,1) = 2 , $\frac{∂^2f}{∂x^2}$ (0,1) = 2 , $\frac{∂^2f}{∂x∂y}$ (0,1) = -1 , $\frac{∂^2f}{∂y^2}$ ...
0
votes
1answer
103 views

The Derivative of a General Linear Map

This question is somewhat abstract compared to the things we've discussed in class, so I'm just making sure I've got the right idea. I'd appreciate any help/suggestions; I'm pretty sure I've got the ...
1
vote
2answers
435 views

Equation of plane that goes for intersection of 2 planes and is perpindicular to another plane

Really don't know what to do here, went to a tutor neither did he. Okay first the problem: Find the equation of the plane that passes through the line of intersection of the planes x − z = 2 and y + ...
3
votes
1answer
101 views

Multivariable Calculus Vector Fields

I have to prove that if $f(x,y,z)=f_{a}(x,y,z)+f_b(x,y,z)+f_c(x,y,z)$ is a conservative vector field and and $g(x,y,z)=g_{a}(x,y,z)+g_b(x,y,z)+g_c(x,y,z)$ is also a conservative vector field, then ...
2
votes
2answers
61 views

transforming a vector from cartesian to spherical and cylindrical co-ordinate system

I know the formula(which i don't know how to copy here but it was in matrix form) for transforming a vector from cartesian system to spherical or cylindrical coordinate system. But, I want to know its ...
0
votes
0answers
23 views

Is this a second derivative of a function of a vector.

Suppose that we have a function f: $f: \mathbb{R}^n \rightarrow \mathbb{R}$. $X = (x_1, x_2, ..., x_n)^T$ is an vector in $\mathbb{R}^n$. I wonder how do they call the following vector: $a_j = ...
2
votes
1answer
42 views

multi-variate normal distribution distance from vector sub-space

let $X\sim {\cal N}(\mu,C)$ be a random variable obeying multi-variate normal distribution in $\mathbb{R}^n$ and $U \subset \mathbb{R}^n$ be a vector space with $\dim(U)=n-1$. What is the probability ...
0
votes
1answer
233 views

How to find plane equation by line and plane that perpendicular to

Find an equation for the plane that is perpendicular to the plane 2x +2y=1 and passes through the line ...
0
votes
1answer
52 views

Conditions for linear independence of extended vector systems

Assume $$g: R^n \times R^m \rightarrow R^n$$ $$h: R^n \times R^m \rightarrow R$$ $$(x,y) \in R^n \times R^m$$ I would like to show that the following vectors are linearly independent: ...
1
vote
2answers
51 views

What is the difference between these two functions?

$$r(x) = \langle x, x^2-1 \rangle$$ $$f(x)=x^2-1$$ Their graph is the same, but one is called vector valued function while the other one is a regular one. I think I'll never get to understand this ...
1
vote
3answers
130 views

what is the relationship between vector spaces and rings?

Can you show me an example to show how vector and scalar multiplication works with rings would be really helpful.
0
votes
1answer
42 views

Finding vector length based on parallell and orthogonal vectors

Do anyone know a simple way of finding the length of vector a in my figure? The known values are $(x_0, y_0), (x_1, y_1), (x_2, y_2), (x_3, y_3)$. (If you look closely, you can see that $f$-vector is ...
0
votes
1answer
53 views

Finding constant multiple of parallel vectors

Given two n-dimensional vectors that are parallel, is there any way to computationally find k such that $$\vec{v}_{1} = k\vec{v}_2$$ without prodding into the components?
6
votes
1answer
192 views

How much can we “cheat” and use vector knowledge in complex analysis?

I'm an engineering-physics student taking a course in complex analysis, and it's a little frustrating, because I see all these connections to vector calculus over the reals (especially as applied to ...
1
vote
1answer
27 views

Tangent vectors as curves equivalence relation

I do not understand the definition of the equivalence relation that is defined on the curves creating a tangent vector space. Let $X$ be any manifold, a point $x \in X$, two curves $\alpha:(-a,a) \to ...
1
vote
0answers
53 views

Vector calculus: Finding a model fit from a 2D grid

First of all, I am a biologist. My knowledge of math is very limited. Therefore, I come here for help, but please take into consideration that I may not be very good at asking the question in a good ...
0
votes
1answer
42 views

Do the paths intersects? If so where

There are two unidentified objects in the sky. The path of the first object is given by $r(t)=\langle t,-t,1-t\rangle $ and the second object's path is $s(t)= \langle t-3,2t,4t\rangle$ Do the paths ...
0
votes
2answers
185 views

How to prove that $f(x,y)=\sqrt{x^2+y^2}$ is continuous in $\mathbb{R}^2$? [duplicate]

Please, I need the demonstration (step by step) of the continuity in $\mathbb{R}^2$ of the function $f(x,y)=\sqrt{x^2+y^2}$. I know that the function is continuous in $\mathbb{R}^2$, but I just don´t ...
2
votes
3answers
80 views

Showing that the product of vector magnitudes is larger than their dot product

QUESTION Show that $$|\mathtt{u} \cdot \mathtt v| \le |\mathtt u||\mathtt v|$$ ATTEMPT Let $ \mathtt {u,v} \in \mathbb R^n$ such that $ \mathtt u = u_1 x_1 + u_2 x_2 + ... + u_n x_n, \mathtt v = ...
2
votes
0answers
137 views

Inner product of two complex vectors?

Given $A \in \mathbb R^{m \times n}$real matrix. If $\langle x,y\rangle =y^{*}x$ for all $x,y\in \mathbb C^{n \times 1}$, can someone help me find the relationship between the following two ...
2
votes
1answer
198 views

Difference between Scalar field and a multivariable Function?

If a scalar field gives out a normal number for every orders pairs what's the difference between it and a function.
2
votes
1answer
95 views

Vector-by-Vector derivative

Could someone please help me out with this derivative? $$ \frac{d}{dx}(xx^T) $$ with both $x$ being vector. Thanks EDIT: I should clarify that the actual state I am taking the derivative is $$ ...
1
vote
1answer
760 views

True Velocity and Heading

An airplane flies at $670$ MPH directly northwest. Wind blows at $70$ MPH from the west (i.e the wind is blowing towards the east). Determine the true velocity and heading of the plane. Steps: ...
1
vote
2answers
178 views

Show that the area vectors for a general $n$-sided closed shape sum to zero

It is possible to show that the sum of the area vectors for a general, closed, $n$-sided figure in $\mathbb{R}^3$ (3-space) is zero. Hint: it may be easiest to consider orientable and non-orientable ...
3
votes
2answers
1k views

Show that the area vectors for a general tetrahedron sum to zero

Using vector addition and multiplication, it is possible to show that the sum of the area vectors for a general closed tetrahedron in $\mathbb{R}^3$ (3-space) is zero. Hint: start by writing down ...
0
votes
2answers
38 views

Curve defined by a vector

http://i.stack.imgur.com/tD4Bn.png I'm studying line integrals with a curve as a vector, but I couldn't understand the 'dr' part. First of all: the curve isn't really a curve, it's like some points ...
2
votes
1answer
112 views

How to approach graphing vector functions?

Let's say I have the following vector function: $\mathbf{r}(t) = t \cos t\,\mathbf{i} + t\,\mathbf{j} + t \sin t\,\mathbf{k}$ What properties of this function will allow me to sketch a curve ...
3
votes
1answer
285 views

Finding the equation of a line entirely defined by a three variable equation.

How can you find the equation of a line that lies completely in a set defined by a three variable equation. For instance, the equation of a line entirely in the set defined by $x^2 + y^2 - z^2 = 1$ ...
0
votes
2answers
197 views

Set of points reachable by the tip of a swinging sticks kinetic energy structure

This is an interesting problem that I thought of myself but I'm racking my brain on it. I recently saw this kinetic energy knick knack in a scene in Iron Man 2: ...
2
votes
0answers
51 views

Coordinate Transform partials question

I wish to go from cartesian to cylindrical coordinates using the chain rule. I see here that $x = rcos(\phi) $ $y = r sin(\phi)$ $r = \sqrt{x^2 + y^2}$ $ \phi = arctan(\frac{y}{x})$ I am ...
1
vote
2answers
652 views

What is the difference between surface area and scalar surface integrals?

What is the difference between the surface area of a paremetrized surface and the scalar surface integral of a function in $\mathbb{R}^3$? Are they not the same thing?
2
votes
1answer
154 views

Use Stokes' Theorem to evaluate integral

Use the stroke theorem to evaluate $$ \int_C{ \vec{F} \cdot \vec{dr}} $$ where C is oriented counterclockwise as viewed from above. $$ \vec{F} = \langle x+y^2, ...
2
votes
1answer
1k views

Line integral vs Arc Length

I am trying to understand when do to line integral and when to do arc length. So I know the formula for arc length varies based on $dx$ or $dy$ like so: $s=\int_a^b \sqrt{1+[f'(x)]^2} \, \mathrm{d} x$ ...
0
votes
2answers
208 views

Integral with vector field in a circle.

Given $$F(x, y) = x^2\mathbf{i} + xy\mathbf{j}$$ $$x^2 + y^2 = 49$$ Find the work done by the force field on a particle that moves once around the circle oriented in the clockwise direction. I've ...
0
votes
1answer
2k views

Parametric equation of line parallel to a plane

The parametric equation of the line is $$x=2t+1, y=3t-1,z=t+2$$ The plane it is parallel to is $$x-by+2bz = 6 $$ My approach so far I know that i need to dot the equation of the normal with the ...
1
vote
1answer
336 views

Basic arc length integration problem

How would you find the arc length of $r(t) = \langle\sqrt{t}, t,t^2\rangle$ for $1\le t\le 4$? This isn't a homework question, I'm just trying to understand how to properly solve a question such as ...
0
votes
1answer
87 views

inequality between entries of the vector and $l_2$ norm of the vector

Let $a=(a_1, \ldots, a_n)$ be a vector in $R^n$. I am wondering for which vectors the following would be true: $$ \|a\|_2^2\geq c \sum_{i\ne j,i,j}a_ia_j, \quad i,j=1, \ldots, n. $$ Here $c>0$ is ...
1
vote
1answer
154 views

Create wind animation

I'm trying to visually illustrate forecast wind speed and direction, the programming is the easy part, the math, I'm fuzzy on. I have a grid of points (lat/lon) , the forecast wind speed and ...