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).
22,852 questions
0
votes
0answers
10 views
How to convert spherical co-ordinates of a vector field to cartesian co-ordinates :
v$(r,φ,θ) = (r cos2 θ)$r$ − (rcosθsinθ)$θ$ + 3r$φ, where r, θ and φ are the unit spherical vectors.
I was trying to calculate the line integral of the function along the path described in the ...
1
vote
1answer
21 views
Existence of a function with $||grad f||>\epsilon$
I want to construct a function $f$ on the unit ball $B$ of $\mathbb{R}^n$, such that it is negative on a closed subset of the boundary $\partial'B\subsetneqq\partial B$, zero on a given point $p\in B$,...
0
votes
1answer
30 views
Understanding the domain of the triple integral for $f(x,y,z)=x+2y+z^2$
So, I am having trouble (again) with the domain for a triple integral of a function, bounded by the paraboloid $2y^2=x$ and the $x+2y+z=4$ and $z=0$ planes
I have tried to guess the bounds for x,y ...
0
votes
2answers
15 views
Application of the Divergence Theorem with change of variable
Let $S$ be the ellipse $\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2 + \left(\frac{z}{c}\right)^2=1,$ with $\vec{n}$ oriented outwards. Compute $\int\!\!\!\int_S \vec{F}\cdot \vec{n}\,dA$ for ...
0
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0answers
13 views
Partial differential Equation uniqueness
Let $\Omega\in\mathbb{R}^{n}$ be a bounded connected open set. I have the following partial differential Equation;
\begin{align}
\nabla\cdot\left(-D(x)\nabla \psi\right)&=F\quad \text{in}\quad \...
0
votes
0answers
12 views
Reference for the multivariate Leibniz rule of many factors
I'm looking for a reference (a book/article) with a formula to
$$
\frac{ \partial ^ k }{ \partial x_1^{k_1} ...
\partial x_n^{k_n} } f_1(x) ... f_m(x) ,
$$
where $k=k_1+...+k_n$, $x=(...
2
votes
0answers
27 views
If $f(x,y)=9-x^2-y^2$ if $x^2+y^2\leq9$ and $f(x,y)=0$ if $x^2+y^2>9$ study what happens at $(3,0)$
If$$f(x,y)=\begin{cases}9-x^2-y^2&\text{if }x^2+y^2\leq9\\0&\text{if }x^2+y^2>9\end{cases}$$study the continuity and existence of partial derivative with respect to $y$ at point $(3,0)$.
...
1
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1answer
39 views
Partial Derivative Disambiguation
There are at least two substantially different meanings to $\frac{\partial}{\partial x}f(x,\ y,\ z(x))$. The $\partial x$ could mean "with respect to $x$ the independent variable," or it could mean "...
3
votes
1answer
25 views
ODE in $\mathbb{R}^n$ defined by the gradient of a function
I'm studying for an exam and I got stuck in this question:
Let $x: I \to \mathbb{R}^n$ be a differentiable parametrized curve (I is an interval) in $\mathbb{R}^n$ and $f: \mathbb{R}^n \to \mathbb{R}$...
1
vote
0answers
23 views
Chain rule for a transformation from $\mathbb{C}$ or $\mathbb{C}^2$ to $\mathbb{R}^2$
Note that, for $z=x+iy\in \mathbb{C},$
$x=\frac{z+ \bar z}{2}\in \mathbb{R}~\hbox{and}~y=\frac{z-\bar z}{2i}=-\frac{i}{2}(z-\bar z)\in \mathbb{R}.$
I was wondering if the following is true:
$$
\frac{\...
0
votes
1answer
38 views
Computing partial derivatives of $f(a,b)= \int_{0}^{1}(ax+b+\frac{1}{1+x^2})^{2}dx$ using chain rule.
Let $f(a,b)= \int_{0}^{1}(ax+b+\frac{1}{1+x^2})^{2}dx$. I want to compute $\frac{\partial{f(a,b)}}{\partial{a}}$ and $\frac{\partial{f}(a,b)}{\partial{b}}$. I was told in the text that $$\frac{\...
0
votes
1answer
19 views
Finding the curve of intersection of a cylinder and cone
I have a cone $x^2 + y^2 -z^2 =0$ and a cylinder $ x^2 +y^2 -2ax =0$. Together they look like this.
If one were to project the intersection onto the xy plane, the curve given by this intersection ...
3
votes
2answers
20 views
Showing a mapping is bijective if and only if a matrix is invertible
Let $\mathbf{A}$ be an $n\times n$ matrix and let $\mathbf{c}$ and
$x_{\star}$ be point in $\mathbb{R}^{n}$. Define the affine mapping
$\mathbf{G} : \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ by
...
0
votes
1answer
23 views
Understanding the domain of the triple integral for $f(x,y,z)=x^2+y^2$
So, I am having trouble with the domain for the triple integral of $f(x,y,z)=x^2+y^2$, bounded by the paraboloid $x^2+y^2=2z$ and the $z=4$ plane
I am currently trying to project it on the XY axis, ...
0
votes
1answer
20 views
Checking saddle point or not - using rules of 'Fundamental Theorem of Calculus'
True or false
For the function, $f(x,y)=\int_{2x}^{-y+2{x^2}}e^{-{t^2}}dt$ $(0,0)$ points is the saddle point.
I can do it the long way by solving the integral first but I believe there is a way to ...
0
votes
0answers
13 views
Proving $\arg$ restricted to an open subset of $S^1$ is smooth
Let $U$ be an open subset of $S^1\subset \mathbb{R}^2$.
Define $\theta:U\to \mathbb{R}$ by $\theta(x,y)=\arg(x+iy)$ where $\arg$ is the principal argument. I want to prove that $\theta$ is smooth (i.e....
3
votes
0answers
10 views
Can we perturb a map to have distinct singular values?
Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \text{GL}_n^{-}$ be smooth. ($\text{GL}_n^{-}$ is the set of $n \times n$ real matrices with negative ...
-2
votes
4answers
60 views
What's “limit doesn't exist” multiplied by limit that equals zero?
OK, we're having a strong discussion just a day before our Calculus exam. The problem's next: To check if this function is continuous:
$$\frac{y^2\,\sin x}{x^2 + y^2}$$ at (0,0).
We get DNE $\cdot 0$, ...
0
votes
1answer
43 views
Find the area between $r=a\cos(\theta)$ and $r=a(1+\cos(\theta))$
So, I have to calculate an integral with a domain limited by two functions: $r=a\cos(\theta)$ and $r=a(1+\cos(\theta))$ , where $a>0$
The issue here is that I cannot wrap my head around what the ...
0
votes
1answer
16 views
How does Green's theorem imply the divergence theorem in the plane?
Both Green's theorem and Stokes' theorem involve the integral of a curl and it is easy to see that Green's theorem is a planar version of Stokes' theorem.
However, the divergence theorem involves the ...
0
votes
0answers
22 views
Prove Jacobian of $f: \mathbb{R}^{2} \to \mathbb{R}^{2}$ with 3 conditionals over $\mathbb{R}^{2}$ is $I_{2 \times 2}$.
If $f: \mathbb{R}^{2} \to \mathbb{R}^{2}$ is given by:
$f(x,y)=
\begin{cases}
(x,y-x^{2}) & if & x^{2} \leq y \\
(x,\frac{y^{2}-x^{2}}{x^{2}}) & if & 0 \leq y \leq x^{...
1
vote
1answer
30 views
Picture flow of ODE
Consider the ODE
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0, \quad x \in \mathbb R^2 \\
\Phi(x,0) = x, \quad x \in \mathbb{R}^2
\end{cases}$$
Suppose that the flow $\Phi$ ...
1
vote
0answers
13 views
Estimating multivariate random walk plus noise model using OLS
I'm currently working on replicating a scientific paper for practice in which they estimate a multivariate random walk plus noise model, apparently using OLS. I have no clue however how they would do ...
0
votes
0answers
7 views
Compute the flux of $F(x,y,z)=(2x,2y,2z)$ through $S=\{(x,y):x^2+y^2=9,0\leq z\leq 5\}$
Compute the flux of $F(x,y,z)=(2x,2y,2z)$ through $S=\{(x,y):x^2+y^2=9,0\leq z\leq 5\}$
So I parametrized $S$ by $\sigma(r,\theta)=(r\cos \theta,r\sin \theta,r^2)$ where $0\leq r\leq \sqrt{5}$ and $0\...
0
votes
0answers
14 views
Will lagrange multiplier method find all stationary points, or just minima and maxima?
Let $f,g : \mathbb{R}^n \to \mathbb{R}$ be smooth functions and $U_c = g^{-1}(c)$ for each $c \in \mathbb{R}$. For each $\lambda \in \mathbb{R}\setminus\{0\}$, consider the equations
$$
\nabla g = 0 \...
2
votes
1answer
19 views
Why do we take magnitude into account when calculating the directional derivative?
Given that the directional derivative is defined formally as:
$$
\nabla_\vec{v}\, f\left(\vec{x}\right) = \lim_{h \to 0} \frac{f\left(\vec{x} + h\vec{v}\right) - f\left(\vec{x}\right)}{h|\vec{v}|}
$$
...
0
votes
0answers
29 views
Triple integral to calculate the volume of pyramid vs it's formula
It is well known the formula to calculate the volume of a pyramid:
$V=\frac {1} {3} bh$, where where $b$ is the area of the base and $h$ the height from the base to the apex.
However I need to ...
0
votes
1answer
26 views
Question about the gradient of a composite function
I am new to calculus and am trying to work out the following question, with no success so far… Any feedback would be great!
Within function $f(x,y)$, variable $y$ is a function of $(x,z)$, in other ...
0
votes
1answer
20 views
Question about gradient in a complex function
I am new to calculus and cannot see the logic of the following question… Any feedback will be really appreciated!
The function $f(x,y,z)$ is differentiable at all points, and satisfies $f(x,y,2x^2+y^...
0
votes
2answers
22 views
Expand the given function in an appropriate cosine or sine series. (Fourier series)
$$f(x) = \begin{cases} x-1, \quad& -\pi < x <0 \\ x+1, & 0 \leq x \leq \pi \end{cases}$$
This is odd so we expand with an odd series
$$b_n = \frac{2}{\pi} \int_{0}^{\pi} (x+1) \sin(nx) ...
1
vote
1answer
36 views
Fourier series confusion
Find the Fourier series of
Question 1
$$
f(x) =\begin{cases}
0&&\text{for $-1 < x < 0$}\\\\
x&&\text{for $0 \leq x \leq 1$}
\end{cases}
$$
and
Question 2
$$f(x) = x + \pi,\;...
3
votes
2answers
27 views
Prove the vector calculus identity $\frac{1}{2}\mathbf{\nabla(\lVert u \rVert ^2) = (u \cdot \nabla)u+u \times (\nabla \times u )}$
My attempt:
Consider the first component of both sides.
$$LHS=\frac{1}{2}\frac{\partial}{\partial x_1}(u_1^2+u_2^2+u_3^2)=u_1 \frac{\partial u_1}{\partial x_1}+u_2 \frac{\partial u_2}{\partial x_1} + ...
0
votes
1answer
38 views
Selling inventory while maximizing the present value of profit
A company owns an inventory of $100$ units of a good. It must sell the entire inventory over the next three periods. The profit function for sales within any one period is
$$\pi(x_t) = 50x_t − ...
0
votes
1answer
25 views
Showing that $f: D_1(0) \subset \mathbb{R}^2 \to \mathbb{R}, f(x) = \frac{1}{1- \|x\| }$ is continuous
I understand that I need to show that for all $x_0 \in D_1(0)$,
$$\lim_{x\to x_0} \frac{1}{1- \|x\| } = \frac{1}{1- \|x_0\| }$$
But I have trouble bounding the denominator,
$$\bigg| \frac{1}{1- \|x\...
0
votes
1answer
37 views
Integration on Differential Forms.
So I know that divergence and curl of a vector field $F$ can be related to a differential form $\alpha$ by div $ F = \star d \star \alpha$ and curl $ \cdot F = \star d \alpha $ , where $\star$ is the ...
11
votes
2answers
154 views
Find $a$ and $b$ for which $\int_{0}^{1}( ax+b+\frac{1}{1+x^{2}} )^{2}\,dx$ takes its minimum possible value.
Calculate for which values $a$ and $b$ the integral
$$\int_{0}^{1} \left( ax+b+\frac{1}{1+x^{2}} \right)^{2}\,dx$$
takes its minimum possible value?
For being honest I'm not sure how to ...
0
votes
0answers
25 views
Why can't these two mappings be bijective?
Let $\phi : \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a continuously differentiable function and define the mapping $\mathbf{F} : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ by
$$\mathbf{F}(x, y) = (\...
1
vote
0answers
41 views
Total differential and overestimation
Consider the function $f(x,y)=10-(x^2+y^2)$ at (1,1) with dx=2 and dy=3
The function value at (1,1) is 8.
Computing the change in the function by taking a first order total differential gives us $-(...
1
vote
1answer
22 views
Double integral to calculate using Polar coordinate system?
I've met a double integral which seems to be calcultated in polar system.
$\iint_D \frac {y} {x^2+y^2} dxdy$ , where $D$ is the region bounded by the following conditions:
$-2y \le x^2+y^2 \le -4y$;
...
0
votes
1answer
24 views
Why is the normal vector different in cartesian coordinates vs. spherical coordinates?
Consider the sphere $x^2+y^2+z^2=1$. Let $\mathbf x(u,v)$ be a parameterization for the sphere.
Say I was trying to find specifically the normal vector given by $$ \frac{\partial \bf x}{\partial u} \...
1
vote
1answer
25 views
Question concerning implicit function theorem
I am studying the implicit function theorem. I have a question about the condition $F_y(x_0,y_0)\neq 0.$
More precisely, let $w=F(x,y)$ be a $C^1$ function on an open rectangle $R=(a,b) \times (c,d)$...
2
votes
1answer
40 views
Find the volume of the solid bounced by the planes $z=0$,$z=y$ and $x^2+y^2=1$
So I do the following: $$\int_{-1}^{1}\int_0^{\sqrt{1-x^2}} \int_0^{y} \,dzdydx$$, but the answer gives me $\frac{2}{3}$, as it graphs a cylinder it should be the half of the half of a cylinder of ...
1
vote
1answer
56 views
Proving that $\lim_{(x,y)\to(0,0)} (x^2+y^2)^{x^2y^2}=1$ using polar coordinates
Am I doing this right? I rewrite the function as follows:
$$(r^2\cos^2\theta+r^2\sin^2\theta)^{r^4\cos^2\theta\sin^2\theta} \stackrel{\text{various trig identities}}{=} r^{\frac{1}{4}r^4\sin^2 2\...
0
votes
0answers
14 views
How to minimize piecewise binary function with absolute value threshold
Let $f: \Bbb R^n \rightarrow \Bbb R$ and $\mathbf{g}:\Bbb R^n \rightarrow \Bbb R^n$. Let $\mathbf{a,b}$ be random variables with values in $\Bbb R^n$.
$$f\mathbf{(a,g(b))} =
\begin{cases}
...
4
votes
1answer
38 views
Exponential decay of the gradient if the function itself and the Laplacian have exponential decay
Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be twice continously differentiable. For $n=1$ one can prove using Taylors formula that we have
$$ \sup_{\vert x \vert \leq R} \vert f'(x) \vert \leq 2\...
1
vote
0answers
51 views
How to evaluate the line integral (checking Stokes Theorem)
Consider the vector field:
$$\vec F = ye^x \hat i + (x^2 + e^x) \hat j + z^2e^z \hat k$$
A closed curve $C$ lies in the plane $x + y + z = 3$, oriented counterclockwise. The parametric ...
1
vote
1answer
22 views
Why curl of a vector field that is proportional to $1/r^2$ equal to $0$?
The curl of the vector field ${\bf F} = (-y {\bf i} + x {\bf j})/(x^2 + y^2)$ is $0$.
I have an intuitive understanding of why the divergence of a radial field that is proportional to 1/r^2 is equal ...
1
vote
1answer
15 views
Evaluate the surface integrals using divergence theorem ${\oint \oint}_S (xy\bar{i} + z^2 \bar{k}) \bar{n} dS$. Help finding domain
Evaluate the surface integrals using divergence theorem
$$
{\oint \oint}_S (xy\bar{i} + z^2 \bar{k}) \bar{n} dS
$$
where S is the surface enclosing the volume in the first octant bounded by the ...
1
vote
1answer
41 views
Use the Stokes's theorem to evaluate the line integral $\oint_c y dx + x dy + (x^2 + y^2 + z^2) dz$
I am using the Stokes's theorem to evaluate the following line integral,
$$
\oint_c y dx + x dy + (x^2 + y^2 + z^2) dz
$$
where $C$ is the curve $x^2 + y^2 = 1, z = xy$ directed clockwise as viewed ...
0
votes
0answers
22 views
Prove that $(\mathbf g\circ\mathbf f)_*=\mathbf g_*\circ\mathbf f_*$ and $(\mathbf g\circ\mathbf f)^*=\mathbf g^*\circ\mathbf f^*$
Let $\mathbf f:\mathbf R^n\rightarrow\mathbf R^m$ and $\mathbf g:\mathbf R^m\rightarrow\mathbf R^k$. I figured out the pull-back part by finding
$$ (\mathbf g\circ\mathbf f)^*(du_1)=d(g_1(f_1(x_1,...,...
