# 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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### Question regarding plotting a Scalar Field that has 3 input variables and 1 output variable

I have a hard time understanding how the equipotential surfaces for a fucntion that has 3 inputs and 1 output is drawn Here ,how is the Contour surface drawn ,is it done by ...
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### Finding the gradient of the restricted function in terms of the gradient of the original function

The following question showed up as part of a proof that I am doing for my research thesis. If we have a differentiable function $f: \mathbb{R}^n \to \mathbb{R}$ and then set $n-d$ coordinates to ...
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### How to find the directional derivative by definition? [closed]

I have the following function: $$f(x,y)=3xy^2+e^{xy}$$ I first calculated it in the normal way, using the unit vector, and the resulting value was $\sqrt2$. Then, I tried calculating it by ...
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### Can the gradient exist for a function of $n + 1$ variables?

For a function of $n + 1$ variables $f(x_0, x_1, x_2, ...x_n)$ can a gradient exist? When I asked my professor this during class he said, "no, at most a gradient will exist for a function of ...
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### Which types of matrices conserve the gradient operator?

Let $M \in \Bbb R^{3 \times 3}$. Let $\Omega\subset \mathbb{R}^3$ be a bounded Lipschitz domain and $u :\Omega \to \mathbb R$ be an infinitely-differentiable scalar field. What conditions should ...
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### Can a field be a gradient of a scalar?

I am trying to answer a question in my electromagnetics book (Cheng). I am given a field A with non-zero curl. It is then asked if A can be expressed as a gradient of a scalar. Note that its not ...
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### How to differentiate 2nd order Taylor expansion of scalar field? [duplicate]

I'm wondering how I can minimize this function with respect to $x$ (not $x_0$). This isn't for homework - I saw them give the answer in the book but they didn't explain how they did it and I'm ...
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### Finding the gradient of $f(x,y)=\max \{|x|,|y| \}$

We have the function $f : \mathbb{R}^2 \to \mathbb{R}$ given by $$f(x,y) := \max \left\{ |x|, |y| \right\}$$ Let $C := \left\{ (x,y) \mid |x|=|y| \right\}$. Given point $(x,y) \notin C$, how can I ...
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### Gradient of a quadratic form with respect to a complex vector

How would I go about calculating the derivative with respect to $x$ of $$Q(x) = x^H A x$$ with $A$ a real matrix (not necessarily symmetric) and $x$ a complex valued vector? Here $(\cdot)^H$ denotes ...
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### Features of a given scalar field [closed]

I have to create a visualization of a scalar field given by the formular: $$f(x,y) = x^3 - 3xy^2$$ I have to represent some features of this scalar field. I plotted the following scalar fieldbut can´t ...
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### Showing $∇$ x $\left(f^2∇f\right)=0$
I am trying to show that $∇$ x $\left(f^2∇f\right)=0$ (the zero vector) I know that from the expansion we get $∇f^2$ x $∇f\:+\:f^2∇$ x $∇f$ From this we can say that $f^2∇$ x $∇f$ = 0 as the curve of ...