Questions tagged [multivariable-calculus]
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).
36,824 questions
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Equivalent definitions of vector-valued Riemann integral
Let $X$ be a Banach space and $[a, b]$ a finite nondegenerate closed interval. I wish to define the Riemann integral of a function $f : [a, b] \to X$. There are two natural definitions that I think ...
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Question about integral of a multivariable function
How should I simplify this expression?
$$g'(t)\cdot \int f(x)\,dx$$
Where $t$ is a constant relative to $x$.
I have a few ideas for what it might be, but I’m new to integrals of functions with ...
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Exact differential eqn. [closed]
To check eaxctness of a diiferential eqn given by M(x,y)dx + N(x,y)dy why we check for dN/dx =dM/dy and doesnot check dM/dx = dN/dy..¿?
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Geometric meaning of double integral when integrating past intersection point
I’m evaluating the double integral
$$\int_{0}^{1/2} \int_{(\sqrt{3})y}^{\sqrt{1-y^{2}}} 1 dx dy$$
The outer limit stops at $y = \frac12$ because the curves
$$x = (\sqrt{3})y$$
$$x = \sqrt{1-y^{2}}$$
...
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Can you express divergence free vector fields as a cross product of gradients?
I know that a divergence free vector field in $R^3$: $A$ as a curl: $\nabla\times B$.
My question is: if we have a vector field $A$ in $R^3$ such that $\nabla\cdot A =0$, does there exist scalar ...
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partial derivative with respect to $x$ of $f(x,y,z)=8-((x-4)(y-3)(z-6))^2$
I am currently in Calc 3 and dealing with minima and maxima of multivariable functions. One of the homework problems my teacher handed out was:
Find the critical points of the function $f(x,y,z)=8-((x-...
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Two functions having the same zero set
If $f$ and $g$ are $C^2$ functions $\mathbb{R}^2\to\mathbb{R}$ having the same zero set $\cal C$.
I want to ask whether
$$\kappa_f:= \frac{\text{Hess}_f(t,t)}{\|\nabla f\|}$$
is equal to (up to a sign)...
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Different (but equivalent) expression of a pullback
Consider the map $\varphi: M \to N$, $x^i$ a coordinate system on $M$ and $x'^i$ a coordinate system on $N$. $\alpha$ is a form.
I was given the "fact" that $$(\varphi^*\alpha)_i(p) =\frac{\...
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Check whether saddle point or not
The question says
Let $f,g: \mathbb{R^2}\to \mathbb{R}$ defined as $f(x,y)=x^2 -\frac{3}{2}xy^2$ and $g(x,y)=4x^4-5x^2y+y^2$ for all $(x,y)\in \mathbb{R^2}$,
consider the following statements and ...
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Doubt about the formulation of the implicit function theorem (and its application to manifolds)
Notation
Let $O \subseteq \mathbb{R}^{n+m}$ be open and denote $\mathbf{x} = \left( x_1,\ldots,x_{n}\right) \in \mathbb{R}^{n} , \mathbf{y} = \left( y_1,\ldots,y_{n}\right) \in \mathbb{R}^{m}$. In ...
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Translation error in Kolmogorov's 1957 Superposition Theorem
Looking through the web, I found a translation of the paper
"On the representation of continuous functions of several variables as superpositions of continuous function of one variable and ...
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Chain rule applied to a multivariable function
Consider the function $f(x(s,t),y(s,t))$. Then, the partial derivative with respect to t is defined as:
$\frac{\partial f}{\partial t} = \frac{\partial f}{\partial x}*\frac{\partial x}{\partial t} + \...
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How to see that the Jacobian is a map between tangent spaces?
At the moment I am learning about manifolds. I am trying to better understand the differential of a smooth map between manifolds.
It is said that "we have to explore the way smooth maps affect ...
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On the existence of points on nD surfaces closest to the origin
Everyone who's gone through the usual calculus sequence has been faced with problems like these:
Find the point(s) on the surface of the ellipsoid $$E = \{(x, y, z) \in \mathbb{R^3} \mid \frac{x^2}{...
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How to calculate $\frac{df}{dg}$? [closed]
Consider functions $f(x(t),y(t))$ and $g(x(t),y(t))$.
I want to understand how to calculate
$\frac{df}{dg}$.
Since I had no idea on how to approach this I considered two examples:
Example 1:
First I ...
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Find all values of $k$ for which the inequality holds for all $x, y$ in the interval $(-1, 1)$
Find all real numbers $k$ for which the inequality
$\frac{1}{x^2 + y^2 + 1} \leq k$
holds for all $x, y$ satisfying $-1 < x < 1$ and $-1 < y < 1$.
My initial thought was to find the ...
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Question regarding proof of Conservative Vector Fields Theorem
This is based on the proof given for theorem 7(Conservative Fields) from Marsden, Vector Calculus fifth edition.
Let $C$ be a simple and oriented curve, parameterized by $c$, that joins the points $(0,...
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Proving the maximum of plane in $\Bbb{R}^3$ bounded by triangle region is at the corners?
I recently did this problem:
Find the absolute minimum and absolute maximum of the plane $z=f(x,y)=17-4x+9y$
on the closed triangular region with vertices $(0,0),(9,0),(9,10)$. List the minimum/...
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Finding the points on a cone where the tangent plane contains the line formed by intersecting two given planes
There are several resources (especially Calculus books) that talk about finding tangent planes to a given (level surface) with certain prescribed conditions. I was trying to make a general question ...
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Why can't we use implicit differentiation to derive determinant of jacobian
I believe this question might also be related to $dA=dxdy$ or $dx\wedge dy$, but since during integration we take the absolute value of the determinant of the Jacobian, the explanations in those ...
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Does the triangle connecting three points on a smooth surface approach the tangent plane as two of the corners approach the third?
The accepted answer at https://math.stackexchange.com/a/3335229/268333 (with 25 upvotes) says that
in the case of a smooth surface, if points $A$, $B$ and $C$ are on the surface, then plane $ABC$ ...
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If an operator is invariant with respect to 2D rotation, is it also invariant with respect to 3D rotation?
Problem: For a function $f(x,y,z)$ and a rotational change of coordinates $(x,y,z)\to (u,v,w)$, the following relation holds
$$\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+\frac{...
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question about the use of Schwinger Feynman parameters.
Note: This is a crosspost of https://physics.stackexchange.com/questions/860755/use-of-schwinger-feynman-parameters-in-a-complex-integral
I am trying to evaluate the following integral:
$$ -\int \...
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How to find the derivative of $f:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$ defined to be the scalar product?
Problem. Define $f:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$ by $(x,y)\mapsto x\cdot y$, the dot product. Find the derivative of $f$ at some point $(a,b)$.
My issue. If $f:\mathbb{R}^n\to\mathbb{R}^...
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Are induced electric fields relatively localised to changing magnetic fields?
Let $\mathbf{d} \colon \mathbb{R}^3 \to \mathbb{R}^3$ be a divergence-free $C^\infty$ function such that $\mathbf{d}=\mathbf{0}$ outside a bounded subset of $\mathbb{R}^3$. Define $\mathbf{E}\colon \...
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