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).

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6
votes
3answers
75 views

The set of all matrix with rank $n-1$ is a hypersurface.

Prove that the set $M$ of $n\times n$ matrices with rank $n-1$ is a hypersurface in $\mathbb{R}^{n²}$ and find the tangent space at $A=(a_{ij})$ where $a_{ij}=\begin{cases} \delta_{ij} \ \text{if} ...
1
vote
2answers
41 views

Calculating fluxintegral out of the surface $1-x^2-y^2$

I am trying to calculate the flux integral of the vector field $$ \vec{F} = (x,y,1+z) $$ Out of the surface $z=1-x^2-y^2$, $z\geq 0$ Answer : $\frac{5\pi}{2}$ I begin by defining a vector that ...
0
votes
1answer
22 views

Flux integral over triangle

Compute the flux of the field $$ F=(x-y+xy, \, -2x+y, \, xz)$$ over the plane triangle with cornerns in $(1,0,0), \, (0,1,0), \, (0,0,1)$. My method: $\\$ Parametrize the triangle by $r=(x, \, y, ...
0
votes
0answers
14 views

Hadamard's method of descent

I would like to apply the Hadamard's method of descent to calculate the solution of the homogeneous wave equation in one dimension descending from the solution in two dimensions. This is the ivp in ...
0
votes
1answer
23 views

Parameterizing part of sphere

the part of the sphere given by: $$ S = \{ (x,y,z) | x^2+y^2+z^2 = 25, -4 \leq x,y,z \leq 4 \} $$ first Q: I'm not sure if I can apply to this Divergence theorem ? It seem that in order to use it I ...
1
vote
2answers
37 views

Continuity of $f$ and existence of directional derivative of $f$

Let $f: \mathbb R^2 \rightarrow \mathbb R $ be defined as $$ f(x,y) = \begin{cases} \dfrac {xy^2}{x^2+y^4} & x\ne 0 \\\\ 0 & x=0 ~ \end{cases} $$ Let $D_u f(0,0)$ denote the directional ...
0
votes
1answer
28 views

Problem in deducing gradient in spherical coordinates.

I know the differential displacement in spherical coordinate as $$dr \cdot \widehat{r}+ r d\theta\cdot\widehat{\theta} + r\sin\theta d\phi\cdot \widehat{\phi}$$. But I can't figure out how the ...
1
vote
1answer
49 views

Find the interval(s) in which function is dercreasing.

For what values (or intervals) of 'a' it holds $a(a-1)x^{a-2}(x-2)+2a(x+1)^{a-1}<0$, where $x\ge2$. I tried to do it by first derivative test but it again gives almost same type expression which ...
0
votes
0answers
22 views

Abuse of Leibniz notation

How is it 'algebraically' justified that, for $f(t)=(x(t),y(t))$ $$\frac{\partial f}{\partial t} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial t}+\frac{\partial f}{\partial ...
1
vote
1answer
24 views

Continuity of partial derivatives only along their axis?

My main question for which I will give an example right below is whether for a partial derivative to exist at a point (say $\frac{\partial f}{\partial x}$) it is necessary for it to be continuous at ...
1
vote
1answer
19 views

Domain of a two variables function

I want to draw the domain of a two variables function. I did it, but not sure my answer is correct. I can't load images due to being new here, so I will describe what I got, and hopefully you can ...
1
vote
2answers
41 views

Clarifications about the correct way to solve exercises (continuity, partial derivatives, differentiability)

I need some clarifications about the correct way to solve an exercise. I have this function: $$f(x,y)=\frac{(x-1)y^2}{\sin^2\sqrt{(x-1)^2+y^2}}$$ and I have to analyse the existence of partial ...
3
votes
0answers
149 views

integral of product of curve with itself in three dimensional

I have the next problem: Let $\gamma:[0,1]\to R^3$ differentiable curve piecewise, and let $\Delta_r=\{(s,t)\in [0,1]^2| |\gamma(s)-\gamma(t)|<r\}$, i want to know if: ...
3
votes
1answer
45 views

Multivariable calculus chain rule for weak derivatives

Let $g:(0,1) \rightarrow \mathbb R^n$ be absolutely continuous, $F \in W^{1,2} (\mathbb R^n).$ Is it true that a.e. it holds $$ \dfrac{dF(g(t))}{dt} = \nabla F(g(t)) \cdot g'(t) \quad ? $$ What I ...
0
votes
2answers
26 views

Show that acceleration is always perpendicular to the radius vector

I need a little help on this problem: "A particle moves with radius vector $\vec r(t)=acos(\theta)\hat i +asin(\theta)\hat j+bcosh(\omega t)\hat k$ where $\omega$ is a constant and $\theta$ is a ...
1
vote
1answer
25 views

Determinant of bounded matrices with transposes

For $A$ an $n \times k$ matrix, $B$ an $n \times n$ matrix, and $I_n$ the identity matrix in $\mathbb R^n$ (where $n$ is some finite positive integer), I need to show for any $\epsilon>0$, there ...
0
votes
1answer
22 views

is the function $f(x,y)=(x\sin(y),x\cos(y))$ on the given interval injective?

Consider $f:(0,\infty)\times (0,3\pi )\to \mathbb{R}^2$, given by $$f(x,y)=(x\sin(y),x\cos(y)).$$ Is $f$ injective? I have to find $(x,y)\in f:(0,\infty)\times (0,3\pi )$ such that $x\not= y$ but ...
0
votes
2answers
23 views

If the iteration $x^{k+1}=x^k-t_kH_k^{-1}\nabla f(x^k)$ converges superlinearly to a stationary point $x^*\ne x^k$, then $t_k\to 1$

Let $f\in C^2(\mathbb{R}^n)$ $(H_k)_{k\in\mathbb{N}_0}\subseteq\text{GL}_n(\mathbb{R})$ $x^0\in\mathbb{R}^n$ and $$x^{k+1}:=x^k+t_k d^k\;\;\;\text{for }k\in\mathbb{N}_0\tag{1}$$ with ...
0
votes
0answers
19 views

Differential of $\,f(t,g(t,x))\,$ with $\,g(t,x)=x+c\,t\,$ ($\,c\,$ constant)

I'm looking for the differential of the implicit scalar function $\,f=f(t,x+ct)\,$ where $x$ and $t$ are real variables and $\,c\,$ is a real constant, and this is what I have done: Consider ...
1
vote
1answer
26 views

Find the region R for which the sequence converges

Find the region $(x,y) \in R$ for which the following sequence converges $$\left| e^n\frac{(\sqrt{y}-\sqrt{x})^{2n}}{x^n} \right| \to 0$$ I am currently doing number theory research on studying the ...
1
vote
2answers
33 views

Vector Identity Question

I am having some trouble with this question regarding vector diffiriential operators. It seems easy and I am not sure what I am missing. The question: Prove: $$ ...
0
votes
0answers
22 views

More on the implicit function theorem: is this example correct?

I am trying to understand the implicit function theorem so I thought it would be a good idea to work out an example. Please could someone look at this and tell me if it is correct? Consider the ...
1
vote
1answer
31 views

Need some help understanding the condition of the implicit function theorem

The condition for the implicit function theorem is that the (smooth) map $f: \mathbb R^n \to \mathbb R^m$ is locally a (smooth) map of $n-k$ variables if there are locally smooth maps $g_i , i \in ...
0
votes
1answer
27 views

Tricky problem dealing with limits of multiple integration,

When we have that $f_n(x)$ converges to f(x) uniformally in some interval, then the limit of the integrals of $f_n(x)$ is the integral of the limit function - over that same interval, of course. A ...
1
vote
1answer
17 views

Use the chain rule to find the derivative of a multivariable function?

I know that $\frac{dg}{dt} = (2xy , x^2)$ Is $\frac{d}{dt} g(r(t))$ simply equal to $\frac{dg}{dt}$ evaluated at $r(t)$? If so, how would I calculate this? $g(x,y)$ depends only on $x$ and $y$, ...
0
votes
0answers
31 views

Is this change of variable in an integral correct?

I have the linear transformation $\mathbf{z}=F(\mathbf{u})$ and the functions $\gamma$ and $\hat{\gamma}$ which are related by $\hat{\gamma}(\mathbf{u})=\gamma(F(\mathbf{u}))$ where $F$ is a two ...
2
votes
2answers
38 views

Inquiring about change of parameters in double (OR Triple) integrals

It's known that while replacing $x$ and $y$ coordinates in double integration, by another coordinate system $u(x,y)$ and $v(x,y)$, then $$ \int_{a_2}^{b_2} \int_{a_1(y)}^{b_1(y)} f\left(x,y \right) \: ...
0
votes
1answer
20 views

Proving multivariable limit exists where a/c + b/d > 1

Given that $\frac{a}{c} + \frac{b}{d} > 1$, I am attempting to show that $$\lim_{(x,y)\rightarrow(0,0)} \frac{|x|^{a}|y|^{b}}{|x|^{c} + |y|^{d}} = 0$$ My attempt at a solution: Let's assume the ...
2
votes
1answer
41 views

Why does curl($F$)=$0$ $\iff$ $F$ is conservative?

Why is it true that$$\displaystyle curl (\vec{F})=0 \iff \vec{F}$$ is conservative i.e. $$\displaystyle \exists f~s.t~\nabla f=\vec{F}$$
2
votes
1answer
27 views

Definition of “vector fields never have opposite direction”

Good day! As in my other question I am referring to the book "Differential Equations and Dynamical Systems" by Lawrence Perko, chapter 3.12. I have a question regarding Lemma 2: Lemma 2. If $v$ ...
8
votes
1answer
111 views

Continuity of normalized displacement vector for a smooth closed curve

I am currently working on chapter 3.12 of "Differential Equations and Dynamical Systems" by Lawrence Perko. I am stuck on the continuity of the function $g$ in Theorem 3. My work (up to the prove of ...
0
votes
1answer
14 views

Level curves of two similar functions

I was trying to plot the level curves of f(x,y)=xy and f(x,y)=2^(xy). Algebraically, by doing f(x,y)=c I got that both are the function 1/x multiplied by a constant, once it's simply c, and once ...
0
votes
1answer
16 views

intuition behind saddle point

Recently,I'm studying multivariable calculus. One of my friend said to me that the graph of $z=y^2-x^2$ has a saddle point .I don't understand the concept of saddle point with intuition.Can anyone ...
1
vote
1answer
32 views

Why should the gradient of an $n-1$-manifold in $\mathbb R^n$ be nonzero?

I am reading this chapter about manifolds here and the author writes (page 2): There is a very important restriction we impose on this situation. It is motivated by our recognition from p. 2–43 that ...
0
votes
1answer
23 views

Computing the Jacobian for change of variables, differentiate w.r.t. to new or old variables?

If I go from $x,y$ variables to new $u,v$ variables in integration, is the Jacobian then the determinant of the matrix of partials $u_x, u_y, v_x, v_y$ or $x_u, x_v$ and $y_u, y_v$? I think it is the ...
2
votes
1answer
41 views

What kind of derivative is this?

In this question here the equation of the derivative of the projection map $\pi_i : \mathbb R^n \to \mathbb R$ is given by $$\pi_i(X+H)-\pi_i(X)=\textrm{grad}\ \pi_i(X)\cdot H+||H||g(H)$$ What ...
0
votes
2answers
20 views

How to simplify this equation with change of variables,

I have the equation, after completing the square: $$(x+\frac{y}{2})^2 + \frac {3y^2}{4} + z^2 = 1$$ How can I further simplify this equation? I need to find the volume inside of this surface. ...
0
votes
2answers
33 views

Find the volume enclosed by the surface $S := \{(x,y,z): x^2 + xy + y^2 + z^2 = 1\}$

Find the volume enclosed by the surface $$S := \{(x,y,z): x^2 + xy + y^2 + z^2 = 1\}.$$ My attempt was this: I moved the tricky $xy$-term over to the r.h.s. I now have $$x^2+y^2+z^2 = 1-xy,$$ ...
1
vote
1answer
17 views

Infinite roots of a scalar function

I've been struggling with a problem for a while, I have to proove if the following proposition is true or false: Let $f:\mathbb{R^n}\to\mathbb{R}$ be a smooth funcion (i.e $f \in C¹$). Suppose that ...
0
votes
0answers
37 views

Pointwise inequality in $n$ dimensions

Suppose we have the system $h(x)=0$ and $f(x)=0$, where $h, f$ are real, decreasing and for which their first derivatives are negative (i.e. $\frac{\partial h}{\partial x} < 0$, $\frac{\partial ...
0
votes
1answer
34 views

Computing the Jacobian determinant for a change of variables,

Why do we compute the partial derivatives in terms on the new variables and not with respect to the old variables? Can someone say this intuitively, so that I have the best chance of remembering why? ...
1
vote
1answer
49 views

Computing the volume inside a surface S, using a seemingly unrelated result,

Consider the surface $$S = \{(x,y,z): x^2 + xy + y^2 + z^2 = 1\}$$. What is the volume inside S? This is actually part (b) of the question. I'm not sure which approach to take. But part (a) of the ...
1
vote
1answer
57 views

Formula for the gradient of $F(\rho,\phi,z)$

Suppose $F(\rho,\phi,z)$ is continuously differentiable, I am interested in showing that the maximum directional derivative of $F$ at any point is given by ...
0
votes
1answer
20 views

Leibniz rule and a heat problem with homogeneous initial and boundary data

Problem Consider the following heat equation: $$v_{xx} = v_t , v(0,t)=0, v_x(L,t)=0, v(x,0)=0.$$ Furthermore, $I(t)$ is defined by $$I(t)=\int_0^L [v(x,t)]^2 dx.$$ Complete the following: Apply ...
1
vote
2answers
40 views

Quickest way to determine if a vector field is conservative?

Say I have some vector field given by $$\vec{F} (x,y,z)=(zy+\sin x)\hat \imath+(zx-2y)\hat\jmath+(yx-z)\hat k$$ and I need to verify that $\vec F$ is a conservative vector field. What would be the ...
4
votes
1answer
48 views

Solving $\int_{0}^{1}\int_{0}^{1-y} \sin\frac{x-y}{x+y}\mathrm dx\mathrm dy$

How would I go about solving the following double integral? $\int_{0}^{1}\int_{0}^{1-y} \sin\frac{x-y}{x+y}\mathrm dx\mathrm dy$ I am absolutely clueless on what to do with that sine.
-1
votes
1answer
37 views

Finding min and max under constraints

I have a two variable function: $f(x,y)=4x^2-y^2-xy-2x+6y$. I need to find its absolute minimum and maximum under the constraints: $y=4-2x$, $x \geq 0$, and $y \geq-2$. I am not sure how to do it, ...
4
votes
3answers
35 views

Coupled differential equation arising in flow line.

So, I ran (certainly not literally) across these two coupled differential equation given by: $$x'(t)=\left(x(t)\right)^2-\left(y(t)\right)^2 $$ $$ y'(t)=2x(t)y(t)$$ These equation occurred ...
0
votes
0answers
27 views

Decreasing of power function.

Show that $\frac{3}{4}(x-2)x^{-\frac{5}{2}}-(x+1)^{-\frac{3}{2}}<0$, wherer $x\ge0$. I tried by taking differentiation but then expression become more complicated. I also tried by checking the ...
2
votes
0answers
23 views

Is my graph correct?

My problem told me that: Let $S$ be the surface described by the equation in cylindrical coordinates $z = r^2$, and the inequality $0 ≤ z ≤ 4$, oriented such that the unit normal vector points ...