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).

learn more… | top users | synonyms

8
votes
2answers
32 views

$\text{Alt}\,(\phi_1 \otimes \phi_2 \otimes \phi_3)$

How do I write out $\text{Alt}(\phi_1 \otimes \phi_2 \otimes \phi_3)$ for $\phi_1, \phi_2, \phi_3 \in V^*$?
0
votes
1answer
18 views

convert from spherical co-ordinate system to cartesian

I want to convert $ρ=\cos{ϕ}$ to Cartesian system , after conversion my answer is $z=x^2+y^2+z^2$ , but its not a sphere , what have I done wrong?
0
votes
0answers
27 views

Which solution is correct?

This is a direct application of Stokes Theorem. So $\int_\Omega div F\; dv=\int_ {\partial \Omega} F.n \; dS$ $\Rightarrow 30=\int_S F.n \; dS+\int_D F.n \; dS\Rightarrow \int_S F.n ...
0
votes
0answers
28 views

Understanding notation - strange use of the del operator

I'm currently reading a paper with the following notation with the del operator which i have never encountered before: Does $\nabla _m$ just mean $\frac{\delta}{\delta \mathbf m} $ ? Furthermore, I ...
1
vote
0answers
19 views

I am trying to find the component of b in the direction perpendicular to a. I am trying to find an alternative route to this problem. Does this work?

c being the component of b in the direction perpendicular to a. So I used the triangular law regarding vectors. I wish I could draw a picture to make it more clear. But ill try to explain... proj ...
4
votes
2answers
41 views

Line Integrals FT usage on this strange vector field: so what are the exact conditions?

I really tried thousands of things before deciding to ask here. Searched all over the internet for an answer, but failed to find it. Let's get started with the Fundamental Theorem of Line Integrals. ...
2
votes
1answer
36 views

Integral Inequality for Bound on Gradient of Solution to Heat Equation

My overall aim is to show that, for a bounded solution $u(x,t)$ to the heat equation in $\mathbb{R}^n \times [0,T]$ with boundary condition $u(x,0) = f(x)$, $$\max |\nabla u(x,T) | \leq \frac ...
1
vote
1answer
18 views

Having Trouble finding a simplified power series representation.

Partial fractions seemed the most efficient route to take. However, I am having trouble at the end.
3
votes
1answer
35 views

Find the gradient of $f^*(x)=\langle (\nabla f)^{-1}(x),x)\rangle-f( (\nabla f)^{-1}(x))$ for $x \in \mathbb{R^n}$

I am stuck at the following exercise which serves as a preparation for the upcoming exam: Let $U \subset \mathbb{R}^n$ be open and $f \in C^2(U, \mathbb{R})$ such that $\det Hf(x) \neq 0, \forall ...
1
vote
2answers
24 views

Multivariable calculus converting from cartesian coordinates to cylindrical coordinates

How do I convert this from cartesian coordinates to cylindrical coordinates? I am really confused. $$\int_{-2}^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \int_{x^2+y^2}^4 (x) \ dz \ dy\ dx$$ I changed ...
4
votes
2answers
21 views

Conceptual explanation of integral of divergence.

$\textbf{My understanding of divergence:}$ Consider any vector field $\textbf{u}$, then $\operatorname{div}(u) = \nabla \cdot u$. More conceptually, if I place an arbitrarily small sphere around any ...
1
vote
0answers
23 views

Solving for the parametrization

I was wondering when evaluating line curves, and C is given by something such as $y = x^2$, how do you find the parametrization $<t, t^2, 0>$ ? ( I understand how z was found but not so much x ...
0
votes
1answer
26 views

Derivative of a function which is defined as a derivative

I'm new to this kind of stuff so maybe this is a stupid question but I don't even know what to search on the internet. My problem is that: find the derivative of the following function on $\Bbb R^3$ ...
0
votes
2answers
12 views

Parametrization of ellipse and normal vector

Parametrization of ellipse and normal vector $$F = x^2 \mathbf i + 2x\mathbf j + z^2\mathbf k \\ C: \text{ellipse} \implies 4x^2 + y^2 = 4$$ I'm trying to find the normal-vector here. I see that ...
1
vote
0answers
16 views

Surface integral with domain that contains an infinite cone

I'm stuck on the following question: Find $\iint\limits_S {ydS}$ where $S$ is the part of the plane $z=1+y$ that lies inside the cone $z = \sqrt {2({x^2} + {y^2})} $ I tried to parametrize $x,y$ ...
2
votes
2answers
28 views

A valid method of finding limits in two variables functions?

I was wondering if in finding the limit of a two variables function (say, $F(x,y)$), I can choose the path by let $y=f(x)$, then find the limit in the same way of that in one variable functions. For ...
4
votes
4answers
323 views

Are derivatives linear maps?

I am reading Rudin and I am very confused what a derivative is now. I used to think a derivative was just the process of taking the limit like this $$\lim_{h\rightarrow 0} ...
1
vote
0answers
24 views

Definition of differential form

Why the definition of differential form guarantee that when we do integration using differential form, it is the same as the usual Riemann integral (before we introduce the concept of differential ...
1
vote
2answers
30 views

Evaluating $\int_{-5}^5\int_{-5}^5-\frac{3}{2}|x+y|-\frac{3}{2}|x-y|+15\,\mathrm{d}x\,\mathrm{d}y$

I'm always having the wrong result from the following: $$ \int_{-5}^5\int_{-5}^5-\frac{3}{2}|x+y|-\frac{3}{2}|x-y|+15\,\mathrm{d}x\,\mathrm{d}y $$ I would really appreciate some guidance on how to go ...
2
votes
1answer
19 views

Surface integral over parabolic cylinder that lies inside another cylinder

To be precise, I'm given the following: Find $\iint_K {xdS}$ over the part of parabolic cylinder $z = \frac{{{x^2}}}{2}$ that lies inside the first octant part of the cylinder $x^2+y^2=1$. In ...
2
votes
0answers
37 views

Area of triangle and uncertainty estimation

Heron's formula states that if a plane triangle has sides $a,b{\text{ and }}c$, then its area is given by $A = \sqrt {s(s - a)(s - b)(s - c)} $, where $s = \frac{1}{2} \cdot (a + b + c)$ is half the ...
1
vote
1answer
27 views

Evaluate a double integral bounded by two circles

Evaluate the integral $\iint_R y \ dR$ where $D$ is a region between the circles $x^2+y^2=2x$ and $x^2+y^2=4$ and on the first quadrant. Is my answer true? ...
2
votes
1answer
10 views

Continuity of the integral as a function of the domain

Let $f: \mathbb{R}^n \to \mathbb{R}$ be integrable. Let $C \subset \mathbb{R}^n$ be measurable. Is $$ r \mapsto \int_{rC} f \, \mathrm{d} \mu, $$ where $rC = \left\{ rc \: \middle| \: c \in C ...
0
votes
1answer
43 views

Finding a pair of Orthogonal Vectors

Want: Pair of orthogonal vectors in $R^4$ that are also orthogonal to the vector (1,1,-2,3) My attempt at a solution: I got stuck...
0
votes
1answer
23 views

Let W be the collection of all 2 by 2 symmetric matrices. Describe the orthogonal complement of W. (please)

A matrix is symmetric if $A^T$=A And the standard basis for symmetric matrices is [a,b], [b c] written as rows of a 2x2 matrix (sorry don't know how to make a matrix on this site). My question: How ...
1
vote
1answer
21 views

Why $[D_{f}(x)]^{-1}$ is continuous on $\Omega$?

Let $\Omega$ be an open set in $\mathbb{R}^n,$ and $f:\Omega\rightarrow \mathbb{R}^n $ be in $C^{1}(\Omega)$. Why: If $\forall x\in\Omega$, we have det $D_{f}(x)\ne0$, then $[D_{f}(x)]^{-1}$ is a ...
2
votes
0answers
22 views

Not getting surface integrals

I have this problem from homework: Integrate the given problem over the given surface. $H(x,y,z)=x^2 \sqrt{5-4z}$ over the parabolic dome $z = 1-x^2-y^2, x \ge 0$ I used this formula from my book ...
2
votes
3answers
73 views

Derivative with respect to $x + t$

I am reading through Princeton's lectures in analysis and I am on the 10th page of the first book on Fourier series. In analyzing the wave equation, they state that $\xi = x + t $ and $\eta = x -t$ ...
2
votes
0answers
28 views

What is the volume inside $S$, which is the surface given by the level set $\{ (x,y,z): x^2 + xy + y^2 + z^2 =1 \}$?

The solution given uses a linear algebraic argument that doesn't seem very instructive -- and may not even be correct, I think. We notice from the equation, that the surface is a quadratic form, ...
0
votes
0answers
32 views

Area of mobious strip.

I want to find area of Mobious strip. I found parameterization $\displaystyle x(s,t) = \left(1 + \frac{t}{2} \cos \frac{s}{2} \right) \cos s$ $\displaystyle y(s,t) = \left(1 + \frac{t}{2} \cos ...
1
vote
3answers
28 views

Show that for a gradient system $\bf\dot x= f(x)$, $\frac{\partial f_i}{\partial x_j}-\frac{\partial f_j}{\partial x_i}=0$ for $1 \leq i, j \leq d$ [duplicate]

The dynamical system ${\bf \dot x} = {\bf f}({\bf x})$ is called a gradient system if there exists a function $V({\bf x})$ such that $$ {\bf f}({\bf x}) = - \nabla V({\bf x}) $$ Show that if ...
0
votes
1answer
13 views

is $f$ necessarily not injective in a neighbourhood of $p$?

Let $f:\mathbb R ^3 \rightarrow \mathbb R^3$ such that $f\in C^1$. Assume there is a point $p\in \mathbb R^3$ such that $rank(Df(p))=2$ (where $Df$ is the differential of $f$) is $f$ necessarily not ...
-1
votes
0answers
16 views

Find multiple integral equation solution

$\int_{\theta=0}^{\pi}\int_{\phi=0}^{2\pi}((v\cos\theta\sin\phi+v')^2+(v\sin\theta\sin\phi)^2+(v\cos\phi)^2)\rho r^2\sin\theta d\theta d\phi$ Can you solve this equation please I use symbolab but I ...
0
votes
1answer
28 views

A multivariate function with bounded partial derivatives is Lipschitz

I'm curious if I've done this correctly -- please offer suggestions/corrections if not! I'm new to working in $\Bbb R^n$ so clear insights would be appreciated. The problem: Let $f:\Bbb R^2 \to ...
3
votes
2answers
115 views

Meaning of “dS” in flux integrals?

In a general flux integral of a level surface, of the form $$\iint{\mathbf{F}\bullet d\mathbf{S}}$$ what exactly does $d\mathbf{S}$ represent? I have seen both $d\mathbf{S} = \mathbf{\hat N}dS = ...
0
votes
1answer
12 views

Hessian determinants and meanings

Suppose $f : \mathbb{R}^2 \mapsto \mathbb{R}$ is a $C^2$ map. Can the determinant of the Hessian matrix of $f$ at the same point be different for different bases of $\mathbb{R}^2$ ? What about eigen ...
2
votes
0answers
116 views

Confused about trying to find the correct spherical co-ordinates for this tricky triple integral

I'm having trouble trying to figure out how to change the limits of integration to spherical co-ordinates in this particular question. I was wondering if someone would kindly be able to assist me in ...
0
votes
2answers
31 views

Prove that if v is orthogonal to u, then it is orthogonal to any scalar multiple of u.

I never understand where to start with proofs, but whenever I see them done I understand them. My attempt: For this one could I just use the property of inner products to prove this? That being ...
0
votes
2answers
27 views

Prove that if u and v are vectors in $\mathbb{R}^n$, then $\langle u,v\rangle =1/4\|u+v\|^2-(1/4\|u-v\|^2)$

I seem to always have troubles when starting proofs. My professor said that the proofs he gave us today are mostly one line proofs, but I just don't know where to start with this one. What I've ...
-2
votes
1answer
29 views

An inner product on $M_(2x2)$ is defind by <A,B>=trace($A^T$B). Verify that for any matrices A,B, and C in $M_(2x2)$ the following holds: [closed]

An inner product on $M_{2 \times 2} $ is defind by =trace($A^T$B). Verify that for any matrices A,B, and C in $M_{ 2 \times 2}$ the following holds: < A+B , C > = < A , C > + < B , C >
-1
votes
1answer
51 views

Finding a pair of orthogonal vectors in $R^4$

Find a pair of orthogonal vectors in $R^4$ that are also orthogonal to the vector (1,1,-2,3). What i have tried so far:
1
vote
0answers
34 views

Find a vector field G such that F = curl(G), Given F = …

Given $F = (xe^y, -x \cos z, -ze^y)$, I find that $\operatorname{div}(F) = 0$ thus there should exist a vector field $G$ such that $\operatorname{curl}(G) = F$. I run into issues finding the solution. ...
0
votes
0answers
20 views

Locally injective function is globally injective [duplicate]

Let $f:\mathbb R\to \mathbb R$ be a continuous: Is the next statement true? If $f$ is locally injective for every real $x$ then $f$ is globally injective in $\mathbb R$ I think this theorem is true: ...
0
votes
1answer
22 views

Integral setup, vector calculus

The trajectory of an airplane is given by the parabola y = −x(x − 10)/10 where x and y are measured in km. Set-up an integral (with bounds) to calculate the amount of fuel burned by this airplane on a ...
3
votes
0answers
56 views

Looking for a rigorous treatment of improper multiple Riemann integrals

I'm studying undergraduate-level differential and integral calculus and have recently come across the topic of improper Riemann integrals. I'm familiar with the concept for single-variable functions, ...
0
votes
1answer
13 views

The determinant of f is not invertible when f is zero when the norm of the function is constant.

Let $f:U\subset \mathbb{R^n}\rightarrow \mathbb{R}^n$ differentiable on the open $U$. If $|f(x)|$ is constant, then $Df(a)$ is not invertible for every $a\in U$. How can I prove that?
1
vote
2answers
28 views

Finding both max and min points by Lagrangian

Using the method of the Lagrange multipliers, find the maximum and minimum values of the function $$f(x,y,z) = x^2y^2z^2$$ where $(x,y,z)$ is on the sphere $$x^2 +y^2 +z^2 = r^2$$ So ...
1
vote
1answer
62 views

Find $\int \limits_0^1 \int \limits_x^1 \arctan \bigg(\frac yx \bigg) \, \, \, dx \, \, dy$

Find $$\int \limits_0^1 \int \limits_x^1 \arctan \bigg(\frac yx \bigg)dx \, \, dy$$ So obviously using cylindrical is the way to go to give $\theta r$ inside the integral (after considering the ...
0
votes
2answers
37 views

Finding volume of a cone given density

Let $C$ be the solid cone with the boundary surfaces $x^2 +y^2 = z^2$ and $z = 0$. The density of the solid at point $(x,y,z)$ is $z$. Find the volume of the solid using the integrals in both the ...
0
votes
1answer
25 views

if $g(t)=f(\frac {\cos (t)}t,\frac {\sin (t)}t)$ is a monotonic increasing function for $t>0$ then $\nabla f(0,0)=(0,0)$

Let $f:\mathbb R^2 \rightarrow \mathbb R $ be in $C^1$. suppose that $g(t)=f(\frac {\cos (t)}t,\frac {\sin (t)}t)$ is a monotonic increasing function for $t>0$. prove that $\nabla f(0,0)=(0,0)$. I ...