# 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.

359 questions
Filter by
Sorted by
Tagged with
32 views

### The function is $F(x,y,z) = x^2y^2z$. Curve C is the intersection of the surfaces $z=2-x^2-y^2$ and $z=\sqrt{(x^2+y^2)}$. Calculate $\oint F ds$.

The function is $F(x,y,z) = x^2y^2z$. Curve C is the intersection of the surfaces $z=2-x^2-y^2$ and $z=\sqrt{(x^2+y^2)}$. Calculate $\oint_C F ds$. Attempt: Firstly, to define the term inside the ...
59 views

174 views

### Gradient of $X \mapsto \mbox{tr} \left(BXCX^TB^TBXCX^TB^T\right)$

Let us assume that \begin{equation} f(X)=\mbox{tr}\left(XCX^TXCX^T\right), \end{equation} in which $C\in\mathbb{R}^{r\times r}$ is a symmetric matrix, and $X \in \mathbb{R}^{r'\times r}$. From the ...
83 views

From Boyd & Vandenberghe's Convex Optimization: $\textsf{Example 2.7}\;\;$ The positive semidefinite cone $\mathbf{S}_+^n$ can be expressed as $$\bigcap_{z\not=0} \{ X \in \mathbf{S}^n \mid z^... 0 votes 2 answers 86 views ### Gradient of x \mapsto (w^tx+b)^2 I have a non-zero column vector w \in \mathbb{R^2} and a scalar b \in \mathbb{R}, so it's a function f: \mathbb{R^n} \to \mathbb{R} with this definition: f(x) = (w^tx + b)^2, where x is a ... 0 votes 1 answer 75 views ### Gradient of X \mapsto \frac1{\sqrt{\det (X)}} Let \Bbb S_n^{++} denote the set of n \times n symmetric positive definite matrices over \Bbb R. Let scalar field f : \Bbb S_n^{++} \to \Bbb R be defined by$$ f (X) := \frac1{\sqrt{\det(X)}} $... 0 votes 0 answers 22 views ### Displacement field, vector field or scalar field? Nabla operator applied to displacement field. I have two questions both related. Firstly, is the displacement field a scalar field or a vector field? It is my understanding that because each point in the field has a direction given by ux and uy, ... 0 votes 0 answers 73 views ### Gradient of quadratic scalar field$X \mapsto \mbox{tr} (XAX)$Given$n\times n$matrix$A$, find (with proof) the gradient$\nabla_X \mbox{tr} (XAX)$. The matrix$A$does not depend on$X$. I know that the final answer is of the form: $$\frac{\partial \mbox{tr} ... 0 votes 2 answers 98 views ### Direction of gradient unit vector at r Consider the scalar field in two dimensions f (x,y) = x2 + 4y2. This field has a global minimum at r = (0, 0). Write a computer program in which, starting from the initial point r(0) = (4, 1.5), you ... 2 votes 1 answer 107 views ### Gradient of {\bf x}^\top {\bf A}^{1/2} {\bf x} with respect to \bf A How to calculate the gradient \nabla_{\bf A} \left( {\bf x}^\top {\bf A}^{1/2} {\bf x} \right), where \bf x is N \times 1 column vector and \bf A is N \times N symmetric positive matrix? The ... 0 votes 1 answer 78 views ### Is this gradient inequality correct? Let scalar field f : \mathbb{R}^n \to \mathbb{R} be smooth, i.e., in C^\infty (\mathbb{R}^n), and there exists \bar{x} be a point on the line segment connecting x_1 and x_2. Is the following ... 0 votes 1 answer 55 views ### If a scalar product is not specified on a vector space over an arbitrary field, is there a "standard" scalar product assumed? In Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry, by Jürgen Richter-Gebert we are asked to consider the three-dimensional vector space over an arbitrary field \... 0 votes 2 answers 225 views ### Derivative of the determinant with respect to the vector There is a vector \boldsymbol{v}=[x, y, z]^T and a matrix \boldsymbol{A}(x,y,z). How is the derivative of a determinant of a matrix with respect to a vector calculated, i.e.: \frac{d|\boldsymbol{... 1 vote 2 answers 82 views ### Criterion for globality of extreme point of a scalar field? So, lets say we have \DeclareMathOperator{\int}{int}$$\Omega \subset \mathbb{R}^n$, $f:\Omega\rightarrow \mathbb{R}$ being continuous and partially differentiable on $\int\Omega$. Now there is $x_*\... 0 votes 1 answer 45 views ### Gradient field of scalar field functions The scalar field functions$s$are defined in space by: $$s(x,y) = x^2y^2 + xy - z + C$$ How do I show that all functions have an identical gradient field and how do I calculate it? 6 votes 0 answers 82 views ### If the gradient of two functions are related by a matrix can we say anything about these two functions? Say we have two functions$f, g: \mathbb{R}^n \rightarrow \mathbb{R}$and we have a matrix$M \in \mathbb{R}^{n \times n}$which is constant, i.e. not a function of$x$. Say we know that$\nabla f = M ...
1 vote
Let $f : \Bbb C^{M \times N} \to \Bbb R_0^+$ be defined by $$f(X) := \left\| X X^H - R \right\|_F^2$$ I would like to find the first-order Taylor approximation of $f$. I am familiar with the vector ...