All Questions
Tagged with vector-spaces calculus
342
questions
3
votes
2
answers
71
views
Given $A, B, C, D$ in $Oxyz$ space, find $M \in CD$ such that $MA + MB$ is smallest. Why can't I use AM-GM to solve this?
In the $Oxyz$ space, consider four points $A(-1, 1, 6),$
$B(-3,-2,-4),$ $C(1,2,-1),$ $D(2,-2,0).$ Find $M \in CD$ such that $△MAB$ has the smallest perimeter.
As $AB$ is constant, the task is ...
- 1,426
0
votes
1
answer
28
views
Outward pointing normal Tetrahedron
For this tetrahedron I need to write down the order of the vertices such that the normal vector points out of the tetrahedron.
For the base DAC, I have drawn the normal vector pointing outwards ...
- 261
0
votes
0
answers
46
views
Divergence of a 2d vector.
The formula for calculating the divergence at a point in a vector field is not clear to me.
$$\mathbf{v} = [V_x, V_y]$$
here $V$ is a vector and $V_x$ and $V_y$ are its $x$ and $y$ components.
Actual ...
- 11
2
votes
3
answers
128
views
Difficult Vectors Problem (Calculus & Vectors)
Find parametric equations of a line that intersects line 1 and line 2 at right angles.
Line 1: $[x,y,z] = [4,8,-1] + t[2,3,-4]$ and
Line 2: $[x,y,z] = [7,2,-1] + k[-6,1,2]$.
I've tried solving this ...
- 29
0
votes
1
answer
48
views
Parametrise a Cylinder
I have a cylinder of equation $x^2+y^2=R^2$ where $z$ ranges from $0$ to $h$.
How would I parameterise this? I want to right $r=(R\cos(\theta),R\sin(\theta),)$ but then I can't write $z=z$ because I ...
- 261
0
votes
0
answers
64
views
Puzzling problem about differential calculus on an arbitrary normed vector space
Given two normed vector spaces $E$ and $F$, let $U$ be an open set of $E$, and define the application $T:U\rightarrow L_c(E,F)$ of class $C^1$, where $L_c(E,F)$ is the space of linear and continuous ...
0
votes
2
answers
126
views
Tangent plane to 3 spheres
Given 3 spheres of radius 9 with center at the points $P = (2,1,0)$, $Q = (5,4,0)$ and $R = (3, 1, 2)$. Find the equation, $ax + by + cz = d$, of a plane tangent to the 3 spheres.
I calculated the ...
- 31
1
vote
0
answers
22
views
Quadrant of matrix inverse with special initialization
Let's assume we have a matrix $\boldsymbol{R} \in \mathbb{R}^{n \times n}$ drawn uniformly in $[-1,1]$ and set
$$\begin{aligned} \boldsymbol{A} & =\frac{1}{2+\|\boldsymbol{R}\|} \operatorname{diag}...
- 661
0
votes
2
answers
37
views
Polar Coordinates Double Integral
$I = \int_{0}^{2} \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} x^2y^2 dy dx $
I must evaluate this using polar coordinates.
I have worked out this must be a circle of equation $x^2+y^2 \leq 4$. So the new ...
user1071088
0
votes
0
answers
65
views
The Divergence Theorem - Intuition behind relating divergence with flux
I'm currently a 2nd-year math major in college. After finishing Calculus 3 from the Stewart calculus book, I'm left wanting a deeper intuition on all of the vector calculus topics (Green's Theorem, ...
- 1
0
votes
1
answer
66
views
Intersection of 3 planes with linearly independent normals
In Tom Apostol's Calculus, vol. $1$, exercise $13.17.16$ is:
Prove that three planes whose normals are linearly independent intersect in one and only one point.
We know that every $n$ linearly ...
- 908
0
votes
0
answers
41
views
Visual (geometric) intuition behind a problem involving vector calculus
The following problem is inspired and directly taken from A preconditioner for solving the inner problem of the p-version of the FEM by Sven Beuchler.
I am trying to find a numerical solution of the ...
- 290
0
votes
0
answers
19
views
Gauge-equivalnt fields
Suppose we have
$$V_1 = 0, \ A_1 = \frac{B}{2}(-y,x,0)$$
and
$$V_2 = 0,\ A_2=B(0,x,0)$$
I need to show that the potentials are equivalent and they represent the same magnetic and electric fields ...
- 429
0
votes
0
answers
50
views
Can I normalize a gradient vector where each dimension has a different unit?
Suppose I have a gradient vector
\begin{bmatrix}
x_{1} \ \frac{cm}{s} \\
x_{2}\ \frac{^{\circ}}{s} \\
\end{bmatrix}
where $x_1$ is given in cm per second while $x_2$ is given in ...
0
votes
1
answer
59
views
Why Gauteaux derivative is homogeneous?
According to wikipedia, given a locally convex topological vector space $X$ and a functional $f: X \to \mathbb R$ (let's assume $f$ is a real value function here to make things simpler), and a vector $...
- 336