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.

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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 ...
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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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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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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 ...
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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 ...
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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 ...
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1answer
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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 = ...
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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 ...
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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 ...
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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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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^...
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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 ...
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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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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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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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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-$k$ ...
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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 ...
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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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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 ...
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1answer
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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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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. ...
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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 ...
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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 ...
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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}...
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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
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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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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{...
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1answer
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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 ...
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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., $\sum_i \sum_j A_{i,j}$?...
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1answer
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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 ...
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1answer
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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 ...
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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^...
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4answers
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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-...
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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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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 ...
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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 ...
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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 ...
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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}/(...
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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/...
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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 ...
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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+(...
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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 ...
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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 ...
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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
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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 \...
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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=...
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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)$\...
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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)}{...