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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4
votes
1answer
86 views

If $a\ge b\ge-c\ge0$, is $\sqrt[3]{a-b-c}\ge\sqrt[3]{a}-\sqrt[3]{b}-\sqrt[3]{c}$?

Let $a\ge b\ge-c\ge0$. Is it true that $\sqrt[3]{a-b-c}\ge\sqrt[3]{a}-\sqrt[3]{b}-\sqrt[3]{c}$?
2
votes
1answer
46 views

Why in this proof we get $\alpha \geq 0$?

I've solved the following problem: "Let $u,v \in \mathbb{R}^n$ with $u \neq 0$ be such that $|u+v|=|u|+|v|$ (euclidean norm), show that there's $\alpha \in \mathbb{R}$ with $\alpha \geq 0$ such that ...
3
votes
2answers
46 views

For $a > 0$, $x \in \mathbb{R}^n$, show that $x \mapsto \frac{ax}{\sqrt{a^2 - |x|^2}}$ is smooth

Let $a > 0$, and set $B_a = \{x \in \mathbb{R}^n : |x|^2 < a \}$. Let $\phi : B_a \to \mathbb{R}^n$ be given by $\phi(x) = \frac{ax}{\sqrt{a^2 - |x|^2}}$. Prove that $\phi$ is a diffeomorphism ...
6
votes
1answer
51 views

Is this proof that the vectors are colinear correct?

I was solving the following exercise: "Let $x,y \in \mathbb{R}^n$ be nonzero such that if $z$ is orthogonal to $x$ then $z$ is orthogonal to $y$. Prove that $x$ and $y$ are colinear". My idea was: ...
1
vote
3answers
130 views

How to find the derivative of $F(t) = \int_0^t f(t, x) \, dx$?

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a continuous function. Define $F: \mathbb{R} \to \mathbb{R}$ by, $$ F(t) = \int_0^t f(t, x) \, dx $$ Then, I'm not sure how to get $F'$. If, there are functions ...
1
vote
2answers
107 views

Does $\frac{\partial \dot{r}}{\partial \dot{q}}=\frac{\partial r}{\partial q}?$

Would it considered abuse of notation to say something in the grounds of $$\frac{\partial \dot{r_i}}{\partial \dot{q_j}}=\frac{\partial r_i}{\partial q_j}?$$ It is supposed to follow from ...
1
vote
1answer
536 views

Given an airplane's latitude, longitude, altitude, course, dive angle and speed, how do you find the new latitude, longitude, and altitude?

I have a model for an airplane that includes the aforementioned information. I want to be able to watch this model fly around on a map over time. Finding distance traveled is easy (distance = speed * ...
1
vote
0answers
156 views

Optimization of three variables with a constraint

I have a question puzzling me for a while. I tried using Lagrange multipliers, however it began to get messy as well as the fact that i am new to the method of Lagrange multipliers! The question is ...
1
vote
1answer
72 views

Mathematical question concerning Lagrange multipliers of a Lagrangian

In Lagrangian Mechanics I have in general holonomic constraints of the form $f(q_1,...,q_n,t)=0$ and then I am able to use the method of Lagrange multipliers, where I go from a Lagrangian $L$ to a ...
1
vote
1answer
93 views

product rule for partial derivative conversion

Working problems in Colley's Vector Calculus and I'm refreshing on partial derivatives, in particular the product rule and chain rules. I've come across a problem where we're identifying cartesian ...
2
votes
2answers
131 views

Exam question: Multivariable calculus, differentiation

I've decided to finish my education through completing my last exam (I've been working for 5 years). The exam is in multivariable calculus and I took the classes 6 years ago so I am very rusty. Will ...
1
vote
2answers
218 views

Trouble with gradient intuition

I'm currently learning about gradients, and I thought khanacademy could help me acquiring some intuition. The actual computation is clear to me, however I'm having trouble understand the intuition. ...
3
votes
0answers
213 views

Partial derivative equality

I'm working through Colley's Vector Calculus 3rd edition to reacclimate and introduce myself to all things vectors and differential forms. The following question is stumping me. Suppose $z=f(x,y)$ ...
4
votes
2answers
82 views

What is a directional derivative?

I have encountered this in an online PDE course I'm following but I've never really been exposed to it. I've looked for the 'formal' definitions but I've never really understood any concept by looking ...
1
vote
2answers
36 views

Show that $f:I\to\mathbb{R}^{n^2}$ defined by $f(t)=X(t)^k$ is differentiable

Let $I$ be a interval, $\mathbb{R}^{n^2}$ be the set of all $n\times n$ matrices and $X:I \to\mathbb{R}^{n^2}$ be a differentiable function. Given $k\in\mathbb{N}$, define $f:I\to\mathbb{R}^{n^2}$ by ...
2
votes
1answer
118 views

Integration formula with wedge product.

On $\mathbb{R}^3$, consider a compactly supported $2$-form $$\omega = f_1 \, dx_2 \wedge dx_3 + f_2 \, dx_3 \wedge dx_1 + f_3 \, dx_1 \wedge dx_2.$$ Choose for $S$ the parametrization $h: ...
1
vote
1answer
67 views

Calculate: $\int_A\mbox{ div }v\, d(x,y,z)$

$A:=\left\{(x,y,z)\in\mathbb{R}^3:x^2+y^2\leq 1,0\leq z\leq 1\right\},$ $v\colon\mathbb{R}^3\to\mathbb{R}^3, (x,y,z)\longmapsto (x^3,x^2y,zx^2)$. Calculate $\int_A\mbox{div }v\, ...
0
votes
1answer
91 views

Double Integral: $f(x,y)=x$ if $x=y$, and $f(x,y)=0$ otherwise

Let $\Omega = [0,1] \times [0,1]$. Let $f \colon \Omega \to \mathbb{R}$ be $f(x,y)=x$ if $x=y$, and $f(x,y)=0$ otherwise. I would like to show the integral exists or not using the criterion of Riemann ...
6
votes
0answers
132 views

1-form with positive integral over a path

Given any smooth path $\gamma$ on a manifold $M$, when can we construct a 1-form $\omega \in \Omega^1(M)$ so that $\int_{\gamma} \omega > 0$? It is easy to see that such $\omega$ exists if the path ...
1
vote
0answers
49 views

Calculating $\int_A\langle v,n\rangle\, dS$

Consider $$ A:=\left\{(x,y,z)\in\mathbb{R}^3: x^2+y^2\leq 1, 0\leq z\leq 1\right\},\\ v\colon\mathbb{R}^3\to\mathbb{R}^3, (x,yz)\longmapsto (x^3,x^2y,zx^2). $$ Calculate $$ ...
4
votes
3answers
297 views

A question on generalization of the concept of derivative

I am looking for some material to understand the process of generalization of the concept of derivative. I would not like to just read and apply the definition of the concept of differentiation in ...
2
votes
1answer
161 views

Improper Multiple Integral

I am trying to solve the following exam problem: Let $s$ be a real number. Find the condition under which the improper integral $$I:=\iint_{\mathbb R^2} \frac{dxdy}{(x^2-xy+y + 1)^s}$$ converges, ...
1
vote
2answers
86 views

Simplification or properties of surface integral $\oint|\vec{U}(\vec{r})\times\hat{r}|^2$ over surface of sphere?

Can we simplify the following surface integral in some way, or say anything about the properties it must have? $$ \oint_{\partial\mathbb{B}\left(\vec{0},R\right)} \left|\vec{U}\left(\vec{r}\right) ...
1
vote
2answers
42 views

Find the image of a ring

I'm working on the following problem: Find the image of the ring defined by $4 \lt x^2 + y^2 \lt 16 $ under the mapping $$F(x,y) = \left(\frac{x}{x^2+y^2} , \frac{y}{x^2+y^2}\right)$$ It looks to ...
1
vote
1answer
43 views

If $f:[a,b]\to\mathbb{R}^n$ is a closed differentiable function, then $\exists t\in(a,b)$ such that $\langle f(t),f'(t)\rangle=0$.

Let $f:[a,b]\to\mathbb{R}^n$ be a closed differentiable function. Show that there exists $t\in(a,b)$ such that $\langle f(t),f'(t)\rangle=0$. Could someone help me? Thanks.
6
votes
3answers
410 views

Can a vector field always be written as a cross-product of two other vector fields?

As the title suggests, can any arbitrary conservative vector field, $\bf{F}$ $= \langle P,Q,R \rangle$, where the component functions are functions of $(x,y,z)$, always be written as a cross-product ...
0
votes
1answer
87 views

What does it mean to say that a function is valued in the space of analytic functions?

I am reading some paper and I encountered this statement: ... the coefficients $a_{p,\beta}(t,x)$ [are] of class $C^m$ in $t$, valued in the space of analytic functions of $x$, in a neighborhood ...
0
votes
1answer
44 views

Basic (multivariable) calculus question

I need some help with basic calculus. I asked a question the other day and got a decent answer but there is one step in the answer I just don't understand. Why is ${\partial y_1 \over \partial x_1} $ ...
3
votes
0answers
86 views

Far-field Poynting vector for time varying source [check my math please]

This is a question from classical electrodynamics, but it's the maths of it I need some help in. I have fields $\rho\left(\vec{r},t\right)$, $\vec{J}\left(\vec{r},t\right)$, ...
2
votes
1answer
67 views

Show $\partial _x \int_{(x_0, y_0)}^{(x,y)}P(s,t)ds + Q(s,t)dt = P(x,y)$

There is a theorem from advanced calculus that I'm trying to prove. Suppose $P(x,y)$, $Q(x,y) \in C^2$ on a simply connected domain $D$, and suppose that $P_y = Q_x$ (i.e. $\omega = Pdx + Qdy$ is ...
4
votes
1answer
139 views

Is this always true?

Suppose $\left|x_{1}\right|\ge\left|x_{2}\right|\ge\left|x_{3}\right|$, $\left|y_{1}\right|\ge\left|y_{2}\right|\ge\left|y_{3}\right|$, and ...
1
vote
1answer
469 views

Express the triple integral three different ways

I need to rewrite the integrals like $dx\,dy\,dz$, $dy\,dz\,dx$, and $dz\,dx\,dy$ of the solid bounded by $$x=2, y=2, z=0, x+y-2z=2$$ I do not fully understand how to rewrite the integral different ...
2
votes
2answers
75 views

Use triple integrals to find the volume of…

The solid enclosed by the parabaloid $$x=y^2+z^2$$ and the plane $$x=6$$ I wanted to make sure that I'm setting up the correct integral before I start to integrate it. ...
0
votes
2answers
58 views

how to find this type of definite double integral?

could any one tell me how to find this type of definite double integral? $$\int_{0}^{\infty}\int_{x}^{\infty}{e^{{-y\over2}}\over y}dydx$$ Thank you.
0
votes
1answer
135 views

surface area of a slice of a hemisphere

Imagine a hemisphere on it's base in the horizontal plane (center at the origin). Imagine another plane P which passes through the center (origin) and inclined at alpha to the horizontal (the base of ...
0
votes
1answer
217 views

When is Jacobian invertible?

Let $(f,U)$ be a chart on $M$ a smooth $n$-manifold. I know the inverse function theorem by which $f$ is invertible on $U$ iff its Jacobian is. What about the other direction? Is there any theorem ...
6
votes
3answers
308 views

Newton's method in higher dimensions explained

I'm studying about Newton's method and I get the single dimension case perfectly, but the multidimensional version makes me ask question... In Wikipedia Newton's method in higher dimensions is ...
3
votes
2answers
37 views

Problem checking that a fuction verifies the Laplace equation.

I'm having trouble solving an excersive from a past final that goes like this: Prove that if $u(x,y) \in \mathbb{C}^2$ satisfies $u_{xx} + u_{yy} = 0$ in $\mathbb{R}² - \{(0,0)\} $ then $v(x,y) = u( ...
2
votes
1answer
223 views

How do I define the limits of a double integral in polar coordinates over an annulus?

Evaluate the double integral by re-writing them in polar coordinates: $\displaystyle\iint\limits_{R}\frac{y^2}{x^2}\ dA$, where $R$ is part of the annulus (ring) $9\leq x^2+y^2\leq 25$ lying ...
3
votes
1answer
89 views

Are there any strategy to solve this system of multivariate quadratic equation?

Solve: $$ (\sum_{j=1}^{n}a_{ij}x_{j})(\sum_{j=1}^{n}a_{ij}y_{j})=0, \quad i=1,\cdots, 2n-1\\ \sum_{j=1}^{n}x_{j}y_{j}=0 $$ ,where $a_{ij}$ are known real constants and $x_{j}$ and $y_{j}$ are nonzero ...
4
votes
3answers
907 views

Finding volume using triple integrals.

Use a triple integral to find the volume of the solid: The solid enclosed by the cylinder $$x^2+y^2=9$$ and the planes $$y+z=5$$ and $$z=1$$ This is how I started solving the problem, but the way I ...
3
votes
2answers
1k views

Evaluate the triple integral, tetrahedron

$$\iiint_E x^2dV, \text{where E is the solid tetrahedron with vertices }(0,0,0), (1,0,0), (0,1,0), \text{and (0,0,1)}$$ I need some assistance on setting up the limits. If someone could help me learn ...
3
votes
1answer
59 views

When can an unknown integral be written as an ODE

I am hoping to numerically solve for the unknown function $I(t)$ \begin{align} I(t) = \int_0^tf(x,t)\,dx \end{align} by converting it into one or more ODEs. The function $f(x,t)$ is known. If $f$ ...
0
votes
1answer
42 views

Help clearing doubt about expansion of $\vec i\times(\vec a\times \vec i)$

I have this doubt in vector analysis I need help with. I know that cross product of a vector with itself is a null vector ($\vec a\times \vec a=\vec 0)$ as both point the same direction. Now consider ...
1
vote
2answers
390 views

Prove whether the limit exists where $(x,y)\to(0,0)$, $f(x,y)=\frac{x^4+y^4}{x^3+y^3}$.

Prove whether the limit exists where $(x,y)\to(0,0)$, $f(x,y)=\dfrac{x^4+y^4}{x^3+y^3}$. After doing polar coordination i get the next expression $\displaystyle\lim_{r\to ...
2
votes
3answers
185 views

Maximization of function in 3 variables

If $( x,y,z)$ be the lengths of perpendiculars from any interior point P of a triangle $ABC$ on sides $BC,CA$ and $AB$ respectively then find the minimum value of : $$ x^2+ y^2 + z^2 $$ The sides of ...
1
vote
1answer
124 views

Second Text for Multivariable Calculus

I took a rather disappointing multivariable calculus course this semester -- the (visiting) professor was not demanding at all. We didn't get to what is in most standard calculus III curriculum. What ...
2
votes
1answer
204 views

How to integrate over polar coordinates

Evaluate the following double integral by rewriting it in polar coordinates: $\displaystyle\iint\limits_Dxy\,dA$, where $D$ is the disc with center at the origin and radius 5 I have very little ...
0
votes
1answer
175 views

Maximizing summation given a constraint

I am writing a software-based algorithm to calculate an optimal solution and I am completely stuck. I need to maximize the following summation with respect to x: $ \sum_{k=1}^n {a_k(1+x_k) \over ...
4
votes
3answers
42 views

Help with $\int\limits_{0}^{1}\int\limits_{2x}^{2}x^2\sin(y^4)\,dy\,dx$

I usually do my problems by myself and then check the solution with Wolfram Alpha, but in this situation, it's not helping me at all... I don't know if I got the wrong answer, or if wolfram is using ...