0
votes
2answers
48 views

$l^r \subset l^p$ and is it even a subspace

It is true that for $r<p$ and $r,p \in [1,\infty)$ we have that $l^r \subset l^p$. Is it true that $l^r$ cannot be isomorphic to a subspace of $l^p$?
-1
votes
0answers
12 views

Is this a subspace of R³?

Is the vector (t,3+t,) a subspace of R³? I know for this you have to test the 3 vector space axioms relating to the zero vector, vector addition and scalar multiplication. I'm just having trouble ...
0
votes
2answers
45 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
1answer
32 views

Let f: $\mathbb{R}^2 \mapsto \mathbb{R}^2$ be a linear function, proof about $f$ and directional derivative

I have this $f$ that is linear and I want to show that for any $a,v \in \mathbb{R}^2$ $f(\begin{matrix} a_1 + v_1 \\ a_2 + v_2 \\ \end{matrix})$ = $f(\begin{matrix} a_1 \\ a_2 \\ \end{matrix}) + ...
0
votes
1answer
39 views

Proof about non-compact sets and unbounded functions

Let $A \subset \mathbb{R}^n$ be a non-compact subset. Show that there exists a continuous unbounded function on $A$. I have split this into two parts. Either: $A$ is unbounded but closed, or $A$ ...
1
vote
0answers
13 views

How do I prove that group$B(v) = (x : ||x|| \leq 1) $contains an elipsoid around the axes?

I have no idea on where to begin. Also, I have to prove that the aformentioned set is convex, which I tried by contradicting, and I'm having difficulty with that as well. Any help would be ...
1
vote
1answer
37 views

Discrepancy over matrix exponential

I am trying to compute $\large e^A$ for $A = \left( \begin{array}{ccc} 0 & a \\ 0 & 0 \end{array} \right)$ Using $\large e^A = \sum \limits_{k=0}^\infty \frac{1}{k!} A^k$ Writing out the ...
0
votes
0answers
21 views

Tensor calculus

I want to find the following, $\nabla \cdot (\rho u u)$, where $\rho$ is a scalar and $u$ a vector. I get $$ \partial_\alpha(\rho u_\beta u_\alpha) = \rho u_\beta \partial_\alpha u_\alpha + ...
0
votes
0answers
32 views

Simple question about natural domain of a function and open/closedness of the domain

This is an easy problem, I just want to make sure I understand everything correctly. I want to find the natural domain of the function $\large\sin(\frac{1}{xy})$ and describe whether it is open or ...
0
votes
1answer
33 views

Find the length and direction of $u \times v$ and $v \times u$

So I was given two vectors: $u=-8i- 2j- 4k$, and $v=2i+2j+k$. I was able to figure out the cross product of $u\times v$ which is $6i-12k$, and $v \times u$ which is $-6i+12k$. However, I need help ...
1
vote
1answer
36 views

Calculus - Check if the line is parallel to the plane

Check if $r = (3,0,2) + t(1,-2,2)$ is parallel to the plane $4x + y - z = 10$ Does it lie in the plane? I'm new to vectors and I'm just wondering how would I solve a question like this!
0
votes
0answers
31 views

Evaluate the surface integral

Let S = $\{(x,y,z) \in R^3$ |$ x^2+y^2+z^2=1\}$.Let V = $(v_1,v_2,v_3)$ be solenoidal vector field on $R^3$. Evalute: $$\int_S [x(x+v_1(x,y,z)) + y(y+v_2(x,y,z)) + z(z+v_3(x,y,z))]dS$$
0
votes
1answer
13 views

Given point A(-4,2,3) and B(4,0,1) what conditions is the line: [x,y,z] = [4,0,1] + t[m,n,1] perpendicular to AB?

Then determine a vector equation either in terms of m or n, of the line that satisfies the condition. Attempt: AB = [8,-2,-2] Therefore, the dot product of [8,-2,-2] and [m,n,1] must be zero. ...
0
votes
1answer
31 views

Solve the linear system in two space?

$(x,y) = (-12,-7) + s(8,-5) $ $(x,y) = (2,-1) + t(3,-2)$ attempt Use elimination: $8s - 3t = 14$ $-5s + 2t = 6$ $s = \frac{10}{33}$ find point: $-12 + 8\cdot\frac{10}{33} = -9.575757$ $-7 - ...
2
votes
2answers
40 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
3answers
51 views

Determine the value of k such that the points A(4,-2,6), B(0,1,0), C(1,0,-5) and D(1,k, -2) lie on the same plane.

A(4,-2,6) B(0,1,0) C(1,0,-5) D(1,k, -2) if they lie on the same plane. How can i determine this? How do you know that the points lie on the same plane? Like do i check if they intersect? How ...
0
votes
4answers
771 views

Determine if two straight lines given by parametric equations intersect

Does $[x,y,z] = [4,-3,2] + t[1,8,-3]$ intersect with $[x,y,z] = [1,0,3] + v[4,-5,-9] ?$ Attempt To find out if they intersect or not, should i find if the direction vector are scalar multiples? ...
0
votes
1answer
74 views

How to find $z$-intercept of vector equation

How do I find the $z$-intercept of the vector equation $\left<x,y,z\right> = (6, -2, -3) + t \left<3,-1,-2\right>$ I am so lost, do I set $x$ and $y$ equal to zero, and solve for $z$? I ...
0
votes
2answers
45 views

Given the scalar equation, 8x + 9y = -45, write a vector equation?

scalar equation: 8x + 9y = -45 Attempt: I took the y-intercept and the x-intercept of the scalar equation and got (-5.625, 0) and (0,-5) By subtracting the points i got [5.625, -5] so my vector ...
0
votes
0answers
21 views

Vector representation of a sum

I am trying to solve a minimization problem and I encountered the following sum: $\sum_i^n x_i^4$ I know that $\sum_i^n x_i^2 = x'x$ but I can't figure out what the vector representation is of the ...
1
vote
1answer
99 views

Describe the coordinates of all points that are 10 units from the yz plane.

I don't really understand the y and z coordinates. I know i will have a +/- 10 as the x-value, and the point will be located at the front right above the origin quadrant (is there a better way to ...
0
votes
1answer
48 views

Prove: $1/2(a+b) \times (a - b) = b \times a$ - in three space

Let $\times$ = cross product and $a$ and $b$ have vector signs on them. $a = [a_1, a_2, a_3]$ $b = [b_1, b_2, b_3]$ How would I prove this? Do I expand out cross product? Any help is appreciated!
0
votes
1answer
25 views

Find a coordinate `b` such that 4 points$v_1(1,-1,0), v_2(2, -1, 1), v_3(-1, -1, b), v_4(1, 1- b, 0)$ on the same plain

This question was originaly in my midterm in calculus 2, my professor says I'm wrong (and gave me 0 out of 20 points), when I think I'm right. It's a bit long question, but please read it. If Im ...
0
votes
2answers
67 views

Is there an $x$ that can solve for this vector?

I'm trying to find an $x$ that solves for this:$$\left[\begin{matrix}1\\2\end{matrix}\right]+\left[\begin{matrix}x\\x\end{matrix}\right]=\left[\begin{matrix}2\\1\end{matrix}\right]$$ But I'm not sure ...
2
votes
4answers
136 views

Find a vector that is perpendicular to $u = (9,2)$

Attempt: We know perpendicular vectors have dot product $u \cdot v = 0$ therefore $[9,2] \cdot [x,y]$ = 0 $9x + 2y = 0$ what would I do now? thanks!
0
votes
1answer
33 views

Vector space question with linear forms

If I have vector space of $n$ degree polynomials with real coefficients and $f(0) = 0$ and $f(1) = 0$, and inner product $\langle f,g \rangle = \int_0^1 f'(x) g'(x) dx$, how do I show that there is ...
1
vote
2answers
50 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
1answer
18 views

Why do they use only the output values to plot the points?

I know that a vector-valued function produces two values from one. Consider f(x)=(2x,5x) Why do they use only the output values to plot the points, whereas at a normal function we use the input as ...
0
votes
1answer
14 views

A question regarding lines between points.

On pg.13 of Lang's "Second Course in Calculus", the following is asserted: Let $P=(2,1)$ and $A=(-1,5)$. Then the parametric equation of the line through $P$ and in the direction of $A$ gives us ...
0
votes
0answers
43 views

What is the sub-gradient of $\lVert x- abs( x )\rVert_2^2$ [closed]

Let $x$ be a N dimensional vector, what is the sub-gradient of $\lVert x- abs( x )\rVert_2^2$? where $abs( \cdot )$ is the absolute operator. That is $abs(x) = y$ where $y_i=|x_i|$.
1
vote
0answers
52 views

Calculating Perpendicular Vectors from a single Look Vector

I have one vector that represents a 'look' direction. I need to calculate the 'right' direction and the 'up' direction. Say I have the vector( 0.79, -0.58, 0.188 ), I thought I could compute cross ...
1
vote
2answers
56 views

How is the norm of a partition related to the norm of a vector?

Just finished a course in linear algebra, where the norm of a vector essentially was described as the length of the vector. In calculus, we just started talking about the definite integral of a ...
1
vote
1answer
134 views

Operator curl and gradient

Operator curl $\nabla$ x$(\cdot)$ (x is cross product) working on a $ C^1$ vector field and operator gradient $\nabla(\cdot)$ working on scalar fields. And results of these operators is vector ...
2
votes
3answers
77 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
2answers
126 views

find symmetric line of given two line

I have one question. Suppose that we have two lines given by equations $$y=2x+3$$ $$y=-2x+11$$ I want to find all equations of lines which these two given lines have same distances from them ...
0
votes
1answer
480 views

Vector Field, Scalar Field. Which is meaningful and not?

Let a, b, c be vectors, f(x, y, z) be a scalar field, F(x,y,z) be a vector field. Which of the following expressions are meaningful? I. (a×b)×(c×b) II. |a|(b· c) +|a|(b+c) III. ∇ ×(f F) IV. (∇ ...
0
votes
1answer
101 views

calculate new velocity when the object collides with a vertical wall. [closed]

I'm struggling to solve this question and id be extremely grateful if anyone could lend me a hand. Given the velocity of the object, v, calculate the new velocity, v', when the object collides ...
2
votes
0answers
43 views

How prove this $\frac{|x-z|}{|x-y|}=1+\frac{1}{|x|}\hat{x}\cdot(y-z)+O(1/|x|^2)$

prove that $$\dfrac{|x-z|}{|x-y|}=1+\dfrac{1}{|x|}\hat{x}\cdot(y-z)+O(1/|x|^2)$$ for $|x|\longrightarrow \infty$ where $$\hat{x}=\dfrac{x}{|x|}$$ This problem from book,following is my idea: ...
1
vote
2answers
186 views

Need help understanding the concept of the Jacobian Matrix and its relation to differentiation

As the question suggests, I need help understanding the concepts around the following differentiation of vector-valued functions: 1) The Norm. I understand that Norm to be defined as follows: In the ...
0
votes
1answer
135 views

Prove that D (the differential operator) maps V (a vector space) into V.

I'm quite confused about what "into" means here and, more importantly, how I am supposed to prove that something maps a vector space into (not onto) another vector space. Here's some of the ...
6
votes
3answers
210 views

Learning Math At Home

I want to learn math on my own at home. What is the best method to do so? I would say that I pick things up/grasp concepts pretty fast. I took math until grade 10 in highschool/secondary school and ...
1
vote
5answers
99 views

Can I treat vectors in $\Bbb R^3$ with a component equal to zero as vectors in $\Bbb R^2$?

If I have a vector $(0,4,4)$ and was finding perpendicular unit vectors to this, is it the same case as finding perpendicular unit vectors for the vector $(4,4)$? Meaning a maximum of two possible ...
0
votes
1answer
106 views

Problem related to differential of a map

I dont understand how to solve this problem. Please can you explain the solution clearly? I want to learn how to solve such problems. Thank you
0
votes
1answer
51 views

For which values of s is the function continuous?

since x is a vector x*x becomes $x^2+y^2$. And then what $s$ value should I put?
1
vote
3answers
209 views

3D - derivative of a point's function, is it the tangent?

If I have (for instance) this formula which associates a $(x,y,z)$ point $p$ to each $u,v$ couple (on a 2D surface in 3D): $p=f(u,v)=(u^2+v^2+4,2uv,u^2−v^2) $ and I calculate the $\frac{\partial ...
2
votes
1answer
40 views

least number of planes intersecting a finite number of points in space, but not intersecting origin.

Let $$\mathbb{R}^*=\mathbb{R}-\{0\}$$ and $$N=\{0,...,n\}$$ and $$\mathcal{M}=\{ A\subseteq \mathbb{R}^3\times\mathbb{R}^* \mid (\forall\mathbb{x}\in N^3:\mathbb{x}\ne 0)(\exists(\mathbb{a},d)\in ...
1
vote
0answers
59 views

Changing variables

If there is a change of variables: $$(\vec x(t),t)\to (\vec u=\vec x+\vec a(t),\,\,\,v=t+b)$$ where $b$ is a constant. Suppose I wish to write the following expression in terms of a gradient in ...
0
votes
4answers
511 views

Compact set - prove that supremum is actually maximum

There is a compact set $K$ in $\mathbb{R}^n$. The diameter of this set is defined as follows: $D = \sup\limits_{x,\, y\, \in K}\|x-y\|$. I need to prove there are two vectors $a,b$ in $K$ such ...
2
votes
1answer
811 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: $s=\int_a^b \sqrt{1+[f'(x)]^2} \, \mathrm{d} x$ ...
3
votes
0answers
164 views

Divergence Theorem to prove equality of integrals

I'm trying to wrap my head around this problem - the interplay between $\nabla$ and $\Delta$ is doing my head in. It says to use the divergence theorem. Prove that $$\int_\Omega u \cdot \Delta v\, ...