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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1
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1answer
14 views

Finding potential function of $\vec F =xy^2 \hat i +y x^2 \hat j$

$$\vec F =xy^2 \hat i +y x^2 \hat j$$ My attempt: $$P=U_{x}=xy^2$$ $$Q=U_{y}=x^2y$$ $$\Longrightarrow U=\int P dx=\frac{x^2}{2}y+C(y)$$ $$ U_{y}=\frac{x^2}{2}+C'(y)=Q=x^2y$$ ...
0
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0answers
16 views

curl-free, conservative vector fields in complex analysis

I just verified that for the conjugate of an analytic function $\bar{f}$=u-iv, this conjugate function is curl-free - the Cauchy-Riemann equations forces this to be the case. Then we can consider ...
0
votes
0answers
61 views

Integral $\left(\frac{x+y}{x-y}\right)^4$ using long division? [on hold]

Alright, this one seems silly but I got a nasty answer when trying to break it up using long division. How do you integrate $\displaystyle\int_0^1\int_0^{1-x}\left(\dfrac{x+y}{x-y}\right)^4dydx$? I ...
2
votes
1answer
52 views

Difference between line integrals in complex analysis and real analysis,

The formula in complex analysis is $$\int f(\gamma(t))\cdot(\gamma'(t)dt$$ and the formula in the real variable setting, for a gradient field, is: $$\int F\cdot dr$$ $$=\int f_x\,dx + f_y\,dy + ...
2
votes
1answer
30 views

Trig substitution for integral of $z/(x^2+z^2)$?

So I have an integral $\int_1^4\int_y^4\int_0^z\frac{z}{x^2+z^2}\,dx\,dz\,dy$ but I can't figure out what trig substitution to use on the first step. When I try $z=\cos$ and $x=\sin$, I end up with ...
0
votes
2answers
101 views

Suppose that $f : U \mapsto \mathbb{R}$ has continuous first partial derivatives.

Let U be an open subset of $\mathbb{R}^n$ and C a compact subset of U. Suppose that $f : U \mapsto \mathbb{R}$ has continuous first partial derivatives. Prove that f is Lipschitz on C. Thoughts: Let ...
1
vote
1answer
21 views

Finding the work from $(0,0)\to(1,1)$ of $\vec F(x,y)=xy^2\hat i+yx^2\hat j$

I need to find the work from $(0,0)\to(1,0)\to(1,1)$ of the following vector field:$\vec F(x,y)=xy^2\hat i+yx^2\hat j$ My attempt: $$\oint_{c}\vec F d\vec r=\int_{(0,0)\to (1,0)}\bigg(xy^2\; dx ...
2
votes
2answers
44 views

Evaluate $\oint_{C} e^{-x} \sin y \;dx+e^{-x} \cos y\;dy$

I need to evaluate the following integral using Green's theorem $$\oint_{C} e^{-x} \sin y \;dx+e^{-x} \cos y\;dy$$ $C$: from point $E \to F\to G\to H$ ...
1
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2answers
46 views

What is a real world example of “zero work” done by a conservative vector field?

I have only a high school physics background, so when I study the later parts of multivariable calculus, e.g., Greens, Gauss, and Stokes' theorems, there are some topics that I only know the ...
0
votes
1answer
14 views

How to interpret multiple critical points (from Lagrange multipliers) that all give a maximum value

If I have 6 critical points, 3 of which give the same maximum possible value of a function f(x,y,z), subject to a constraint g=c, is there something more to say about this solution -- or we just ...
2
votes
0answers
48 views

Green's theorem application

Problem Determine all circles $\mathcal C$ on $\mathbb R^2$ such that $$\int_{\mathcal C}-y^2dx+3xdy=6\pi$$ My attempt at a solution If I call $P(x,y)=-y^2$ and $Q(x,y)=3x$, then I can apply ...
1
vote
2answers
525 views

Rotating the gradient

Suppose I have a triangle T in 3dimensional space and i want to rotate it in arbitrary ways. The coordinates for T are given by $f: T_R \in \mathbb{R}^2 \rightarrow T \in \mathbb{R}^3 $ where $T_R$ is ...
0
votes
2answers
67 views

How to differentiate the following interesting vector product?

How do we differentiate the following vector product with respect to $\boldsymbol r$. \begin{equation} \frac{d}{d\boldsymbol r}\bigg[(\boldsymbol \omega \times\boldsymbol r)\cdot (\boldsymbol \omega ...
1
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4answers
28 views

Equation perpendicular to 2 non-parallel planes

Good day sirs! Can you help me with this questions? Find the general equation of the plane: (1) Through $(3,0,-1)$ and perpendicular to each of the planes $x-2y+z=0$ and $x+2y-3z-4=0$ (2) ...
0
votes
2answers
48 views

Solve this set of Lagrange multiplier equations,

I'm trying to solve $$(yz,xz, xy) = (\lambda\frac{2x}{a^2},\lambda\frac{2y}{b^2},\lambda\frac{2z}{c^2})$$ with the constraint equation $$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1$$ ...
8
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0answers
228 views

The Stable Manifold Theorem Applications

Definition: Let $\phi_t(x)$ be the flow of the nonlinear system $x'=f(x)$. The global stable manifold of $x'=f(x)$ at $0$ is defined by: $$W^s(0)=\bigcup_{t\leq 0}\phi_t(S)$$ Where $S$ is a ...
1
vote
1answer
20 views

What is the maximum value of work done by this force field?

An object moves in the force field $F=yz\hat{i}+zx\hat{j}+xy\hat{k}$ starting at the origin and ending at some point $A(\xi,\eta,\zeta)$ that lies on the surface ...
2
votes
1answer
45 views

Show that a closed $1$-form on ${\bf R}^2 - 0$ has the form $\omega=\lambda \,d\theta+dg$

This is Problem 4-30 from Spivak's Calculus on Manifolds: If $\omega$ is a $1$-form on ${\bf R}^2 - 0$ such that $d\omega = 0$, prove that $$\omega = \lambda \,d\theta + dg$$ for some $\lambda ...
1
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0answers
21 views

Critical value example where partial derivative does not exist

Each of the following functions has a critical value where the partial derivatives do not exist. $f(x,y)=(x^2+y^2)^{1/3}$ $f(x,y)=1-\sqrt{x^2+y^2}$ $f(x,y)=3-[(x-1)(y-2)]^{2/3}$ Does anyone have ...
-2
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0answers
12 views

Local exactness implies potential function [on hold]

Let $D$ be a simply connected domain and let $u(x,y), v(x,y)$ be two smooth functions such that $u_y=v_x$ in $D$. (a) Prove that there exists a potential function $\varphi(x,y)$ such that ...
0
votes
2answers
23 views

continuously differentiable multivariable functions

What does it really mean to say a function $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ is continuously differentiable? A function $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuously differentiable if ...
-2
votes
0answers
39 views

Why $ (\cos(\theta) \frac{\partial}{\partial x} + \sin(\theta) \frac{\partial}{\partial y} ) \frac{\partial}{\partial \theta} =0$? [on hold]

Why $ (\cos(\theta) \frac{\partial}{\partial x} + \sin(\theta) \frac{\partial}{\partial y} ) \frac{\partial}{\partial \theta} =0$ ?
1
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2answers
563 views

Maximum/Minimum of Curvature - Ellipse

Find the sum of the maximum and minimum of the curvature of the ellipse: $9(x-1)^2 + y^2 = 9$. Hint( Use the parametrization $x(t) = 1 + cos(t)$) Tried to use parametrization like that, but then ...
1
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1answer
31 views

Limit of weird multivariable function defined by parts

$f(x,y) = \left\{ \begin{array}{ll} 0 & \mbox{if } y \geq x^2 \mbox{ or } y\leq0\\ 1 & \mbox{if } 0<y< x^2 \end{array} \right.$ I want to take the limit as $(x,y)\to (0,0)$ from ...
3
votes
2answers
152 views

Question on Inductive Proof of Implicit Function Theorem

I am struggling with an inductive proof of the implicit function theorem, concretely with the final part of construction of a function, up to this final point everything is perfectly clear to me. ...
-8
votes
0answers
40 views

I need help on problem 54 and 55 (the way to solve this kind of quiz.) [on hold]

I need help in problem $54$ and $55$. How do I solve these kinds of questions? I know how to find gradients and how to find the tangent plane equation and the normal line too.
1
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1answer
510 views

Change of variable (translation) in complex integral

If I have a real integral, e.g. $\int f(x+2) \ dx$, I can substitute $y = x+2$, so $dy = dx$. But if my function is complex, am I still allowed to do this? In which cases I cannot apply a ...
3
votes
1answer
41 views

Can't Finish Double Integral in Polar or Cartesian

Alright, so I'm stuck on what I think should be a simple double integral. It is $\int_0^1\int_{\sqrt x}^1e^{y^3} \, dy \, dx$. This is just the volume between the surface $z=e^{y^3}$ and the area ...
2
votes
0answers
19 views

Determine Critical points in optimisation problem

So I have this problem where I am supposed to calculate the max and min value of a function $f(x,y)=x+2y$ restricted by the disk $x^2+y^2\le 1 $. I have calculated the $df/dx $ and $df/dy$ and they ...
1
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0answers
19 views

Modeling of Multivariate Function of Dependent Variables

In multi-variable calculus, if I write $f(x,y,z)$, it is assumed that $x,y,z$ are independent. I'd like to model a quantity $F$, that is a function of 3 related quantities, $x,y,z$. In fact, $xy=z$. ...
3
votes
1answer
333 views

Prove that hyperspherical coordinates are a diffeomorphism, derive Jacobian

The explicit form for the transformation into hyperspherical coordinates is $$x_1 = r\sin\theta_1 \sin\theta_2 \dotsb \sin \theta_{n-1} \\ x_2 = r\sin\theta_1 \sin\theta_2 \dotsb \cos \theta_{n-1} \\ ...
0
votes
1answer
86 views

Finding the area under the curve $y=3-3\cos(t),x=3t-3\sin(t)$

I need to find the area under the curve $\color{blue}{y=3-3\cos(t),x=3t-3\sin(t)}$ and between $\color{blue}{x=2\pi,x=0\text{, above axis}}$ using $\color{blue}{\text{Green's theorem}}$. My attempt ...
2
votes
1answer
52 views

Finding the area under the cycloid $x=t-\sin (t),\;y=1-cos (t)$

I need to find the area under the cycloid $x=t-\sin (t),\;y=1-cos (t)$ above axis and between $x=0,x=2\pi$ using $\underline{\text{Green's theorem}}$ I found in Wikipedia this evaluation: ...
4
votes
2answers
63 views

Evaluating the line integral $\int_C{F\cdot dr}$ for a particular conservative vector field $F$

So I have this two dimensional vector field: $$F=\langle (1+xy)e^{xy},x^2e^{xy}\rangle$$ How can I tell whether $F$ is conservative or not? And also how do I calculate $\int_C{F\cdot dr}$, where $C$ ...
0
votes
1answer
54 views

How does gradient of a vector point steepest ascent

The derivative of distance function with respect to time give velocity function in single variable calculus. But how does gradient of a multivariable function point steepest ascent? I have been ...
0
votes
1answer
62 views

Evaluate $\oint_{C}xy^2dx+2x^2 dy$

$$\oint_{C}xy^2dx+2x^2y dy$$ triangle:$$C=\{(0,0),(2,2),(2,4)\}$$ My attempt: Using Green's theorem $$\oint_{C}\underbrace{xy^2}_{P}dx+\underbrace{2x^2y}_{Q} dy=\iint\bigg(\frac{\partial ...
0
votes
1answer
95 views

inequality between entries of the vector and $l_2$ norm of the vector

Let $a=(a_1, \ldots, a_n)$ be a vector in $R^n$. I am wondering for which vectors the following would be true: $$ \|a\|_2^2\geq c \sum_{i\ne j,i,j}a_ia_j, \quad i,j=1, \ldots, n. $$ Here $c>0$ is ...
1
vote
4answers
142 views

Prove using $ \varepsilon-\delta $ that $ \lim_{(x,y)\to(1,1)} \frac {x^2+2xy-3y^2}{x^2-y^2} = 2 $

Prove limit using $ \varepsilon-\delta $ definition that: $$ \lim_{(x,y)\to(1,1)} \frac {x^2+2xy-3y^2}{x^2-y^2} = 2 $$ I've been reading quite a lot about how to prove limits; so I want to show ...
0
votes
1answer
28 views

Change of variables when integrating over a triangle

I want to calculate the integral $$ \iint_D(x-y)dxdy $$ where D is the triangle made up of the vertices (0,0), (-2,1) and (-1,3). (Graph) My idea was to do this substitution $$ \begin{equation} ...
-1
votes
1answer
82 views

Line Integral: $\int_C{x^2}\:dy$

How can I calculate $\int_C{x^2}\:dy$ in which $C$ is a line segment from the point $(0,0)$ to $(3,2)$? I am new to line integrals, I am only familiar when given a function and in $ds$. How can I do ...
2
votes
1answer
65 views

Calculate $\int_C{y^2\:ds}$ where $C$ is $x^2+y^2=9$

I need help calculating $\int_C{y^2\:ds}$ where $C$ is $x^2+y^2=9$. So what I first did was convert $C$ into parametric and then I set it up like this: $$\int_0^{2\pi}(3\sin t)^2\sqrt{(-3\sin ...
0
votes
1answer
31 views

Writing line integral as 1-form

If $F: \Bbb R^n \rightarrow \Bbb R^n $ is a vector field and $\phi : [a,b] \rightarrow \Bbb R^n$ is a continously differentiable path we defined the integral of $F$ along $\phi$ as $\int_{\phi} F = ...
0
votes
1answer
58 views

Evaluate $\iint dydx$ on the domain $0\leq r\leq1$, ${\pi}/{3}\leq\theta \leq{2\pi}/{3}$ [duplicate]

I need to evaluate $\displaystyle\iint \color{red}{dydx}\;\;\;,\bigg\{\frac{\pi}{3}\leq\theta \leq\frac{2\pi}{3}\bigg\}\;\;\;\;,0\leq r\leq1$ $\color{blue}{\text{without using polar ...
3
votes
2answers
82 views

Stokes Theorem. Where is my mistake?

Use Stoke's Theorem to prove that the following line integral has the indicated value. $$ \int_\mathscr{C} y \,dx +z\,dy+x\,dz = \pi a^2 \sqrt{3}$$ where $\mathscr{C}$ is the intersection curve ...
0
votes
1answer
44 views

Lipschitz function proof

Statement Let $F(t,X)=A(t)X+b(t)$ with $A(t) \in \mathbb R^{n\times n}$ and $b(t) \in \mathbb R^n$. If the coefficients $a_{ij}(t)$ and $b_i(t)$ are continuous functions of the variable $t$ in a ...
0
votes
1answer
27 views

Green theorem application

Suppose that a simple closed curve $C$ in the $xy$ plane, that bounds a convex domain $D$ containing the origin. The curve is specified by $x=f(\varphi), y=g(\varphi)$ where $0\leq \varphi< 2\pi$ ...
0
votes
1answer
14 views

Calculating min/max of a multivariate function on a region

This video shows an example of how to find the absolute maxima and minima of the function $f=xy+y^2$ at the region $\{(x,y):|x|\leq1,|y|\leq2\}$. I understand why he set $f_x, f_y$ to $0$, checked ...
1
vote
0answers
404 views

Polar Integration over intersection of two circles

Let $C_0$ denote a circle centered at $(0,0)$ with a radius of $r_0$ and let $C_1$ denote a circle of radius $r_1$ centered at a point $(x_1,0)$. Assume that we are given some function, $\phi(r)$ ...
2
votes
1answer
77 views

Doesn't $x^3+2y^3+3z^3=0$ give a surface in $R^3$?

In my last exam on Advanced Calculus (following Spivak's Calculus on Manifolds), I couldn't solve the following question. True or false: the set $S$ in $R^3$ given by $x^3+2y^3+3z^3=0$ is a ...
15
votes
2answers
426 views

How do I generalize the derivatives / integrals from multivariable calc?

$\newcommand{\RR}{\mathbb{R}}$ This is a long post (EDIT, two years later: not as long as my answer), so I'll put the big question right at the top: There's a whole lot of derivative-like and ...