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Use this tag for questions about differential and integral calculus with more than one independent variable. Some related tags are (differential-geometry), (real-analysis), and (differential-equations).

32,781 questions

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Question about definition

I have a very naive question about the definition of the operator $(1+x)\partial_x$ as an operator $C^1 (\mathbb{R})\to C^0(\mathbb{R})$. Is it $A u = (1+x)\partial_xu$ in the sense that we have $Au(-... user avatar Jacques Mardot

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A linear change of variables in a PDE, basic help

Consider the PDE to be solved for $f$, and for a given vector field $b:\mathbb{R}^n\to\mathbb{R}^n$ $$ \partial_t f(t,x)=\text{div}(b(x) f(t,x)),~\forall~t,x \in \mathbb{R}_+\times \mathbb{R}^n, $$ ... user avatar trenkoir viske

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table of parametric surfaces with differentials

Can you send me a free opensource document best would be a pdf, that lists all the common surfaces (sphere cilinder torus cone etc) of parameters and they normal vectors calculated at once. That would ... user avatar konto

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Calculate the total derivative

Calculate directly the total derivative (without using partial derivatives) of the function $f(x_1,x_2)=x_1^2-10x_2$. You may wish to use the fact that $\sqrt{x^2+y^2}\geq\frac{x+y}{2}.$ $$$$ I have ... user avatar Techlover

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Prove $f(x,y) = x^4+x^2y +y^2$ has a minimum at $(0,0)$. [closed]

Prove $f(x,y) = x^4+x^2y +y^2$ has a minimum at $(0,0)$. I tried using the second partial derivative test but the results were inconclusive, any help will be useful . user avatar hopeless.cantor

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Intersection between algebraic curves and its asymptotes

I have been studying the intersection of algebraic curves and asymptotes from the book Taneja, H. C. (2010). Advanced Engineering Mathematics. According to the author, If an Nth degree algebraic ... user avatar Vinny_94'

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Baby Rudin theorem 11.42

There are the definitions which we need for the proof of the theorem : There is the theorem: If ${f_n}$ is a Cauchy sequence in $\mathscr L^2(\mu)$ , then there exists a function $f$ $\in$ $\mathscr ... user avatar JohnNash

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Differentiability of $g(x,y) := f \left(\sqrt{x^{2} + y^{2}}\right)$

Suppose the function $f : \Bbb R \to \Bbb R$ is continuously differentiable and define another function $g$ as $$g(x,y) := f \left(\sqrt{x^{2} + y^{2}}\right)$$ Under what condition is $g$ ... user avatar krishna2016

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Finding a volume using a double integral

Problem: Find the volume of the solid whose base is the region in the xy-plane that is bounded by the parabola $y = 4 - x^2$ and the line $y = 3x$, while the top of the solid is bounded by the plane $... user avatar Bob

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Can the Jacobian change sign when integrating by substitution?

I'm studying calculus with the textbook "Calculus" by James Stewart. I was curious about the sign of the Jacobian when a change of variables occurs. Inside the text's boundary, there doesn't ... user avatar Sungbin Moon

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Difficulty computing volume enclosed by parametrized surface $x=\frac{t^2}{1+t^3},y=\frac {t\cos\phi}{1+t^3},z=\frac {t\sin\phi}{1+t^3} $

Compute the volume of the solid $V$ enclosed by the surface $x(t,\varphi)=\frac{t^2}{1+t^3},y(t,\varphi)=\frac{t}{1+t^3}\cos\varphi,z(t,\varphi)=\frac{t}{1+t^3}\sin\varphi, t\in[0,+\infty),\varphi\in[... user avatar Invisible

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Calculate total derivative directly.

Calculate directly (not via partial differentiation) the total derivative of the function $f(x_1,x_2)=x_1^2-10x_2.$ You may wish to use the fact that $\sqrt{x^2+y^2}\geq\frac{x+y}{2}.$ $$$$ For the ... user avatar Techlover

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Calculating integral of function over group blocked by lines

I cant seem to find an online tool to help calculate these kind of integrals in order to confirm weather my solution is correct/incorrect, the problem is such: Let $Q$ be the group representing the ... user avatar Aishgadol

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Why we do care about open/closed regions in math - specifically in multivariable calculus? [closed]

I am studying multivariable calculus and I have encountered the idea of an open/unbounded vs closed/bounded region. I understand this is similar to the concept of interval notation where $(5,5)$ does ... user avatar John

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Finding Curvature, is $K=\frac{|\vec{s}'(t) \times \vec{s}''(t)|}{||\vec{s}'(t)||^3}$ the right formula to use?

In this lesson, Grant/Khan taught that $K=\frac{|\vec{s}'(t) \times \vec{s}''(t)}{||\vec{s}'(t)||^3}$: = If the formula is right, is it generalisable to all dimensions? I applied it to $\vec{s}(t)= (\... user avatar nvs0000

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