Questions tagged [scalar-fields]
A scalar field is a function of the type $X\to \Bbb R$, where $X$ may be an open set in $\mathbb R^n$ or more generally a smooth manifold.
142
questions
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Functions defined by the second derivative
I am looking for classes of differentiable functions that are characterized by their second derivative.
For example:
convex functions: function $f$ is convex iff its Hessian, $\nabla^2 f$, is ...
3
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2answers
89 views
Calculate the gradient of a linear scalar field [duplicate]
I am trying to calculate the following gradient
$$\nabla_{\mathbf{X}} \left( \mathbf{a}^{T} \mathbf{X} \mathbf{a} \right)$$
where I am using the convention that $\mathbf{a}$ is a column vector. I am ...
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0answers
17 views
Finding irrotational and scalar potential [closed]
Show that the vector $F$ is irrotational and hence find it's scalar potentialenter image description here
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2answers
33 views
Positive definite gradient shift?
Let $f: \mathbb{R}^n \to \mathbb{R}$. Let $M$ be positive definite. Is it always true that $f(x)$ increases in the $M \nabla f$ direction, where $\nabla f$ is the direction of steepest ascent (the ...
1
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3answers
58 views
How to differentiate $g(X)=\operatorname{tr}\left(X^{-1}\right)$? [duplicate]
Let $X$ be a square invertible $n \times n$ matrix. Calculate the derivative of the following function with respect to X.
$$
g(X)=\operatorname{tr}\left(X^{-1}\right)
$$
I'm stumped with this. As when ...
0
votes
3answers
75 views
Matrix derivative of $\mathrm{tr}((I+X^{-1})^{-1})$
I'm trying to calculate the derivative of $\mathrm{tr}((I+X^{-1})^{-1})$ with respect to $X$. By some sort of a chain rule, I believe this should be $X^{-1}(I+X^{-1})^{-2}X^{-1}$. However, I'm having ...
1
vote
1answer
22 views
How to write the expression for the following vectors? (ratio problem)
The diagram shows a parallelogram $ABCD$. $E$ is such a point on $CD$ such that $BD:EB=1:3$. Write expressions for these vectors.
(a) DC
(b) CD
(c) AC
(d) AE
(e) DE
I solved the first three: $DC = ...
-2
votes
1answer
47 views
Arc length (line integral of a scalar field) [closed]
The given arc
I tried and got 0.. The integral I have tried:
$\cos\left(t\right)\sin\left(t\right)\left(\cos^2\left(t\right)-\sin^2\left(t\right)\right)-2\cos^3\left(t\right)\sin\left(t\right)+t$
The ...
0
votes
1answer
40 views
How to compute $\frac{\partial}{\partial X}tr(BXX^tA)$?
I know that $\frac{\partial}{\partial X}tr(BXX^t) = BX + B^tX$ according to the matrix cookbook equation 109.
However, I need to calculate $\frac{\partial}{\partial X}tr(BXX^tA)$. Is there a simple ...
0
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1answer
68 views
Can a discontinuous vector field be conservative?
I consider a closed curve within a 2-dimensional vector field, passing through two points $A$ and $B$. Going clockwise, along the path from $A$ to $B$ the field is constant $\vec{v}(\vec{x}) = \vec{k}$...
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1answer
46 views
Derivative of $\text{Tr}[B X^T A X^{-1}]$
Let $A, B, X \in \mathbb{R}^{n \times n}$ and assume that $X^{-1}$ exists. Derive $\frac{\partial K}{\partial X}$ where
$K(X)= \text{Tr}[B X^T A X^{-1}]$
I have tried the following so far ($U = B X^...
0
votes
1answer
29 views
Why is the directional derivative not defined for $v_1=0$?
I have some trouble understanding how my prof. got to this conclusion.
I'm asked to find the directional derivative at point $\zeta=(0,0)$ of $$\left\{
\begin{array}{c} f(x,y)=\frac{x+xy}{\sqrt{x^2+...
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1answer
27 views
Scalar fields proof
How to prove that for every two scalar fields $u(x,y,z)$ and $v(x,y,z)$ the identity above holds true?
I guess it says the Laplacian of the dot product of two scalar fields equals the Laplacian of ...
0
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1answer
23 views
How to write a function that describes a thin spherical shell?
How can we define a 3D function where for all input values of $(x, y, z)$ for $f(x,y,z) = 0$ or approximately close to zero except for where the location of a spherical shell is located.
The ...
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votes
2answers
53 views
Difference between Directional Derivative x Chain Rule for Scalar Fields
could you guys help me out with an issue I am having.
What's the difference between the "Directional Derivative" and "Chain Rule for Scalar Fields"? In meaning and the formulae?
I don't know if I ...
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0answers
22 views
help me find the value of a when u=(a,-4) and v=(2,6) are parallel vectors.
u=(a,-4) and v=(2,6) are parallel vectors. to find the value of a, I let -4=6k and the find the value of k=-2/3. I multiplied 2, the x component of vector v, with the value of k to get the value of a. ...
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0answers
53 views
Solving linear system over the reals modulo $\pi$
I would like to solve the system of equations
$$\sum\limits_{i \in S_m}{} x_{i} \equiv b_m \quad (\textrm{mod } 2\pi)$$
in $n$ real-valued variables $x_1, \dots, x_n$, where the $S_m$'s are length-$k$ ...
0
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1answer
18 views
Laplacian of two scalar fields
I am trying to find $$\nabla^2 (gh)$$
so far I have $$\nabla^2 (gh) = \nabla\cdot(\nabla(gh)) = \nabla\cdot(g\nabla h+h\nabla g) = \nabla\cdot(g\nabla h)+\nabla\cdot(h\nabla g)$$
I am not sure the ...
0
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2answers
37 views
Index notation to prove vector identity
How can I use index notation to prove this identity? I have not been able to find any good resources on using index notation.
\begin{equation}
\nabla (fg)= f\nabla g + g \nabla f
\end{equation}
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1answer
32 views
How is a set $\mathbb{R}$ a field if for a=0$\in \mathbb{R}$, $\nexists a^{-1}$ such that $a\cdot a^{-1}$=1
I was reading a chapter concerning the definition and examples of fields.
It states that examples of some known fields include $F=\mathbb{R}, \mathbb{Q}, \mathbb{C}$
One of the 9 properties of a ...
0
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1answer
24 views
Find all first-order partial derivative of the given function defined on $R^n$.
$$f(x)=\sum_{i=1}^n \sum_{j=1}^n a_{ij} x_i x_j$$ given $a_{ij}=a_{ji}$
my atempt.
let $g(t)=f(x+te_k)$ where $e_k=$the k-th unite coordinate vector. Then computed $g'(0)=f'(x;e_k)=\sum_{i=1}...
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0answers
84 views
Tangent/Perpendicular lines to a point on a contour line
I don't understand how to find lines that are tangential or perpendicular to a the point (1,1) on the scalar field z=x^2 +4y^2. I have taken the grad operator and plugged the coordinates 1,1 into it. ...
1
vote
2answers
122 views
Derivative of coefficients of characteristic polynomial
Let $ X(t) $ be a square matrix of dimension $ n $. The Jacobi formula expresses the derivative of the determinant of $X(t)$ in terms of the derivative of the matrix itself. Is there an analogous ...
1
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1answer
27 views
Apollonios equation holds iff the norm is induced by a scalar product
Let $(E, \vert \vert \cdot \vert \vert )$ be a normed space. The norm is induced by a scalar product iff in $(E, \vert \vert \cdot \vert \vert )$ the Apollonios equation holds.
The Apollonios ...
1
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1answer
24 views
Sum of scalar products is finite dimensional
I'm trying to prove: for $f \in C^1[0, 2\pi]$ such that $f(0)=f(2\pi)$ show that $\sum_{k=-\infty}^{\infty}\vert \langle f, e_k \rangle \vert < \infty$, where $e_k(t)=e^{-ikt}$ for $k \in \mathbb{Z}...
0
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1answer
21 views
Covergence of sequences in a pre-Hilbert space
I have to prove or disprove the following statement:
Let $(x_n)_n$ be a sequence in a pre-hilbert space $\mathbb{H}$, $x \in \mathbb{H}$. Then $(x_n)_n$ converges to $x$ iff
$lim_{n \rightarrow \...
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1answer
24 views
Does Scalar Field imply it is time-invarant
I am writing an article on data acquisition pertaining to terrain mapping. I want to know if I can collectively call mapping of such static surfaces, scalar fields? So in otherwords does stating ...
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0answers
21 views
what is wrong with following line integrals in a scalar field?
Let $f(x,y) = 1$
Evaluate $ \int_C^ \! f(x,y) \, \mathrm{d}s \,\,$ where $C$ is straight-line segment from $(1,1,1)$ to $(0,0,0)$.
Using $(t,t),$ $1 \leqslant t \leqslant 0 $
$\qquad\qquad$ $\mathrm{...
0
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1answer
47 views
How to check if dot product calculation is true [closed]
I've written a python code that calculates the dot product of two x,y,z points converted from lan/lon points (out of three lat/lon points)
The formula I'm using.
The angle I'm trying to get.
My ...
0
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2answers
115 views
Derivative of summation across all elements of a matrix
Maybe a trivial question but my linear algebra / calculus is not very strong at the moment.
How do I take the derivative wrt a matrix of a summation across all indices, i.e., $\sum_i \sum_j A_{i,j}$?...
0
votes
1answer
61 views
Prove that $ \text {(null}~T^*)^\perp \subseteq \text{range}~T $ [duplicate]
Let $T \in L(V,W)$ where $L(V,W)$ denotes the set of linear maps from $V$ to $W$. Prove that $ \text {(null}~T^*)^\perp \subseteq \text{range}~T $ where $T^*$ is the adjoint operator ( not related to ...
0
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1answer
25 views
Gradient vector of scalar field
Calculate the gradient vector of the following scalar field:
$\Phi(r)=\cos(ar)$ where $r$ is the 3-dimensional position (vector) and $a$ is a constant vector.
So gradient means that I need to ...
0
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0answers
23 views
Expression of time derivative of a scalar
I have been trying to solve this question for some time but still have not figured it out.
Show that given a constant matrix $M$ and any time-varying vector $x$, the time derivative of the scalar $x^...
0
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4answers
129 views
Gradients, Directional Derivatives and Change in Scalar Functions
In single variable scalar function $\ f(x)\ $the sign of the derivative can tell you whether the function is increasing or decreasing at the point. I was trying to find an analogous concept in multi-...
0
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3answers
85 views
Gradient of $f(x)=(a^T x)(b^T x)$
How do you find the gradient of $f(x)=(a^T x)(b^T x)$ where $a$, $b$, and $x$ are $n$-dimensional vectors?
So, far I tried by taking a derivative with chain rule:
$$ D(f(x)) = D[(a^Tx)(b^Tx)] = (a^...
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2answers
31 views
Why in $GF(7)$ there are no nonzero elements satisfying $a^2 + b^2 = 0$?
Why in $GF(7)$ there are no nonzero elements satisfying $a^2 + b^2 = 0$? how can I know this by a quick method without calculations? Is there a theorem or something that can help here?
$GF(7)$ is the ...
1
vote
1answer
30 views
Distribution of multiplication over addition in a field.
I am studying Linear algebra from a book called "The Linear Algebra a Beginning Graduate Student Ought to Know " by Jonathan S. Golan.
And in the definition of a field the book mentioned that the ...
1
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1answer
49 views
If scalar field $f(x,y,z)$ is differentiable then the set of point satisfying $f(x,y,z)=c$ ($c$ is constant) is smooth surface?
Let $f(x,y,z), x,y,z\in \mathbb{R}$ is a scalar field. Could you prove that if $f$ differentiable then the set of points $(x,y,z)$ sastifying $f(x,y,z)=c$ ($c$ is constant) is a smooth surface?
The ...
0
votes
2answers
58 views
Elaboration of a vector space over $GF(2)$ with symmetric difference as addition operation.
The example given below is from Golan's "Linear Algebra":
But I have some questions regarding it.
My questions are:
1- What is the meaning of $GF(2)$ in this example, does it mean $\mathbb{Z}/(...
2
votes
1answer
49 views
Direction of least change of a scalar/vector function
The gradient of a scalar/vector function gives the vector/tensor of greatest change. I am looking for the inverse concept, which gives me the direction of least change.
Inverting the gradient vector/...
1
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2answers
82 views
Gradient of scalar field $a^T X^{-1} b$
During the derivation of GDA as generative algorithm, I am stuck at how to take the gradient
$$\nabla_X \left( a^TX^{-1}b \right)$$
where $a, b$ are column vectors independent of $X$.
I have tried ...
0
votes
3answers
70 views
Proof that $\| u \|=\| v\|\iff\langle u+v,u-v\rangle=0$
Let $(\cdot,\cdot):\mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R}^n$ be any map that fulfills the properties
$u,v,w\in\mathbb{R}^n;\; \lambda\in\mathbb{R};\; \langle u;v\rangle:=(u_1v_1)+\cdots+(...
1
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2answers
409 views
The conjugate of a scalar is the same scalar in a matrix-scalar multiplication?
I am reading the book, Applied Linear Algebra and Matrix Analysis.
When I was doing the exercise of Section 2.4, Page 97, I thought the equation is right as followed:
$(cA)^{*} = cA^{*}$
But answer ...
2
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0answers
29 views
Under what conditions are partial derivatives continuous?
For any continuous scalar and vector fields which doesn't contain any singular points, under what conditions are their partial derivatives continuous?
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1answer
150 views
Matrix derivative of $Tr(A\log(X))$
I'm trying to work out the derivative of $Tr(A\log(X))$ with respect to $X$. Assume $X$ is positive so the $\log$ is well defined. I know that
$$Tr(A\log(X)) = A^\dagger: \log(X)$$
but what I ...
1
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1answer
29 views
Why is this notation equal to its transpose?
In my econometrics textbook, I have this step which is not clear to me:
\begin{align} S &= e'e \\ &= (y-W\beta)'(y-W\beta) \\ &= \underbrace{y'}_{1\times T} \ \underbrace{y}_{T\times 1} ...
2
votes
2answers
84 views
Derivation of $\frac{\partial}{\partial A} \left( y^T A x \right) = y x^T$ [duplicate]
I would like to see a detailed, step-by-step derivation of the following identity
$$\frac{\partial}{\partial A} \left( y^T A x \right) = y x^T$$
where $x, y \in \mathbb R^n$ and $A \in \mathbb R^{n \...
0
votes
2answers
45 views
If $\vec{\nabla} \times \langle P,Q,Q \rangle=\vec{0}$ iff $Pdx+Qdy+Rdz$ is exact differential form.
If $\vec{\nabla} \times \langle P,Q,R\rangle=\vec{0}$ iff $Pdx+Qdy+Rdz$ is an exact differential form.
My attempt:-
If $Pdx+Qdy+Rdz$ is an exact differential form. Then there exists $U(x,y,z): dU=...
1
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0answers
49 views
2 types of Line integrals in scalar field
Consider the line integral,
$\int _ c$f(x,y)$\vec dr$ , where f(x,y) is a scalar field, and it is evaluvated on a curve c . After integration we get a vector let it be $\vec I$ .
$\int _ c$f(x,y)$\...
-1
votes
1answer
53 views
How does the $Tr(nX^{n-1})$ become $n(X^{n-1})^T$?
For a unstructured square complex matrix X. Show that
$\frac{\partial Tr(X^n)}{\partial X}=n(X^{n-1})^T $ and $\frac{\partial Tr(X^n)}{\partial X^*}=n(X^{n-1})^T$.
I think $\frac{\partial Tr(X^n)}{...
