All Questions
Tagged with vector-spaces multivariable-calculus
241
questions
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Applying vector decomposition multiple times and RH orthonormal bases
I want to show that I can write any 3D vector $v$ in components with respect to the right handed orthonormal basis $\{e_1, e_2, e_3\}$ (i.e. three perpendicular unit vectors $\{e_1, e_2, e_3\}$ such ...
- 139
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1
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17
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Taylor Expansion of a vector-valued function with 2 vectors as input
Let a function $f(x,u): \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^n$.
I wonder how to expand it around $(x_n, u_n)$.
For the time being, keeping it only up to the first order is enough
...
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1
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28
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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
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46
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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
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48
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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
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43
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Deriving the hessian of a composition of vector-valued functions
I am running in circles trying to derive the formula for the hessian of a composition of (real-valued) vector functions, and can't figure out where I'm going wrong.
Suppose we have (smooth-enough) ...
1
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116
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Dependency of the Hessian on the inner product
Question: the gradient of a function $f:E\rightarrow\mathbb{R}$, with $E$ being a vector space equipped with an inner product $\langle\cdot,\cdot\rangle$, depends on the choice of the inner product ...
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2
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37
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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
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1
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79
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Verification proof of a separation theorem like in $\mathbb{R}^n$
I would like to prove the following theorem
Let $C$ be a non empty closed convex subset of $\mathbb{R}^n$ that does not contains the null vector and $X_0\in\mathbb{R}^n\setminus C$. Then there exists ...
- 1,381
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41
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What does it mean to say a vector is any set of three components that transforms in the same manner as a displacement when you change coordinates?
In Chapter 1, "Vector Analysis" of Griffith's Electrodynamics he says at some point
a vector is any set of three components that transforms in the same
manner as a displacement when you ...
- 5,021
2
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1
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48
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Prove triangle inequality on $\hat{N}(x,y) = \sqrt{x^2 + xy + y^2}$
I'm having problems to demonstrate triangle inequality on the above function. So far I've tried:
\begin{align}
\hat{N}(u+v) &= \hat{N}((u_1, u_2)+ (v_1,v_2))\\
&= \hat{N}((u_1+v_1, u_2+v_2))\\
...
- 47
1
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29
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Infinite Integrals and "Moments" of a Random Vector
I'm trying to understand the concept of "moments" of random vector, and what this means for (potentially) infinite integrals. Here is an example
Suppose we have a random vector $\alpha$ ...
- 280
2
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1
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40
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Is one-sided differentiable inverse sufficient to conclude equal dimensions?
Let $\Omega$ be an open subset of $\mathbb R^n$ and $\Upsilon$ be of $\mathbb R^m$. Let $f\colon \Omega\to \Upsilon$. Then this answer shows that if there exists a point $c\in\Omega$ where $f$ is ...
- 4,119
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Average value of an integral over a vector space
How do we find the average value of an integral over a vector space for example $\Bbb R^2$ so normally to find the average value of an integral over a region you find $$\frac{\displaystyle\iint_R f(x,...
2
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1
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140
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Uniqueness of high-dimensional derivative [Zorich's book]
I was reading the definition of the high-dimensional derivative from Zorich's book and I'd like to ask a question about the uniqueness of high-dimensional derivative.
Definition 1. A function $f:E\to ...
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