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

0
votes
0answers
18 views

How to find out if point is local Maximizer or local Minimizer ? Lagrangian is given

The Lagrangian is: $L(x,\lambda) = x_1x_2-2x_1-\lambda (x_1^2-x_2^2)$ Taking the derivatives and setting it equal to zero gives: $x_2-2\lambda x_1-2=0$ $x_1+2\lambda x_2=0$ $x_1^2-x_2^2=0$ The ...
-1
votes
4answers
34 views

Length of Spiral in a plane

Problem Take a positive constant real number $c$. Draw a rough sketch and find the length of the spiral in the plane given by $(x(t),y(t))=(e^{-ct}\cos(t),e^{-ct}\sin(t))$ for $0\leq t<\infty$. ...
1
vote
2answers
46 views

Math Subject GRE 1268 Problem 64 Flux of Vector Field

What is the value of the flux of the vector field F, defined on $R^3$ by $F(x,y,z) = (x,y,z)$ through the surface $z=\sqrt{1-x^2-y^2}$ oriented with upward-pointing normal vector field? (A) $0$ (B) ...
2
votes
5answers
31 views

Finding a general solution to a differential equation, using the integration factor method

Use the method of integrating factor to solve the linear ODE $$ y' + 2xy = e^{−x^2}.$$ And verify your answer I can solve the ODE as a linear equation (mulitply both sides, subsititute, reverse ...
1
vote
1answer
45 views

Multivariable Calculus: Manifolds

Problem Let $M$ be the set of all points $(x,y) \in \mathbb{R}^2$ satisfying the equation $$xy^3 + \frac{x^4}{4} + \frac{y^4}{4} = 1 $$ Prove that $M$ is a manifold. What is the dimension of $M$? ...
0
votes
0answers
23 views

Find solution of the initial value problem

Suppose a bar of length $\ell$ is attached to a rigid hub of radius $r$, which together are rotating at a constant angular speed of $\omega$ radians per second. The other end of the bar has a ...
1
vote
1answer
31 views

Good reference on higher dimensional derivatives?

I've spent several months now periodically scouring the internet for a comprehensive overview of an introduction to higher dimensional derivatives. I've already read baby Rudin's section on the ...
-1
votes
1answer
33 views

An exercise in Boothby's book

In pages 39,41 Boothby's book(An introduction to Differentiable Manifolds By William M. Boothby), we see that: I guess that the answer is yes. How can I prove this excersise(chapter II, section 5 ...
1
vote
0answers
24 views

Surface integral on unit circle

Let $S$ be the unit sphere in $\mathbb{R}^3$ and write $F(x)=\nabla V(x)$ where $V(x)=1/|x|$ Evaluate $$\iint_S F\cdot n dS$$ Without using divergence theorem, we can evaluate it straightforwardly, ...
2
votes
1answer
18 views

Let $f$ be differentiable at every point of some open ball $B(a)$ in $\mathbb R^n$ and $f(x)\le f(a) , \forall x \in B(a)$ , then prove $D_k f(a)=0$.

If $f:\mathbb R^n \to \mathbb R$ is a function differentiable at every point of some open ball $B(a)$ with center $a\in \mathbb R^n$ and $f(x)\le f(a) , \forall x \in B(a)$ , then how to show that all ...
0
votes
2answers
14 views

$f'(x;y)=0$ for every $x$ in an open convex set and for every vector $y$ ; then to show $f$ is constant on $S$

Let $f:\mathbb R^n \to \mathbb R$ be a map , $S$ be an open convex set in $\mathbb R^n$ such that for every $x \in S$ and $y \in \mathbb R^n$ , $f'(x;y)$ exists and equals $0$ ; then how to show that ...
2
votes
3answers
38 views

Substitution to solve an initial value problem

By using the substitution $y(x) = v(x)x$, how can I solve the initial value problem $$ \frac{dy}{dx} = \frac{x^2+y^2}{xy - x^2},\quad y(1)=1 $$ And also keep my answer in the form $g(x,y)= 4e^{-1} ...
1
vote
1answer
30 views

Diffeomorphism between Euclidean space

How does one show that if $f:U\rightarrow V$ is a diffeomorphism between open sets $U\subset\mathbb{R}^m$ and $V\subset\mathbb{R}^n$ then $m=n$? Here is some working: For $u\in U$ let $v=f(u)\in V$. ...
1
vote
0answers
32 views

Application of Stoke's Theorem

Edit: I think I misunderstood the problem. Upon reading my textbook again, I think what they mean by $F(x,y,z)=<yz,2xz,e^{xy}>$ ; C is the circle $x^2+y^2=16, z=5$ is just literally a ...
0
votes
1answer
30 views

Stokes' Theorem - The normal vector

Stokes' theorem says: $$\oint_cFdr = \int\int_S curl F dS = \int\int_S curl F \cdot n \, dS$$ Where $F$ is a vector field on $\mathbb{R}^3$. My question is what do I take $n$ to be? If we ...
1
vote
1answer
40 views

Absolute Min and Max of $f(x, y)=x^2+4y^2-2x^2y+4$ Using Partial Derivatives

Consider this problem: Find the absolute minimum and absolute maximum of $f(x, y)=x^2+4y^2-2x^2y+4$ on the rectangle given by $-1\leq x\leq1$ and $-1\leq y\leq1$ I solved this problem using ...
0
votes
2answers
39 views

If a vector field is conservative then is it path independent?

I am studying vector calculus and I am having trouble with the idea of path independence. Is it necessarily true that if $F=(P,Q)$ (a vector field in $\Bbb R^2$) is conservative, then $\oint \limits ...
2
votes
2answers
70 views

Vector Field Conceptual Question

Given that: $$F = \langle yz-2xy^2, axz-2x^2y+z, xy+y \rangle$$ in which $a$ is some constant. Now, for what $a$ would make the vector field of $F$ conservative? Why is there only one, or are there ...
1
vote
0answers
45 views

Under what conditions is this true: $\lim_{r \to 0} \frac{1}{r} \int_{0}^{2\pi} f(r,x) dx = 2\pi f(0,0)$

I will like to know under what hypothesis the following is true, and maybe a sketch of the proof. I saw it in a solution of an exercise. In this exercise, $f$ was harmonic, but I don't know if that is ...
1
vote
2answers
52 views

Evaluating Line Integrals using Green's Theorem

I am currently learning about Green's Theorem, Curl and Divergence, and I came across a problem: Given a two dimensional vector field: $$ F=\langle e^{\sin{x}}+y^2, x^2+y^2 \rangle$$ And then I am ...
1
vote
1answer
21 views

Does given point satisfy FONC?

minimize $4x_1^2+2x_2^2-4x_1x_2-8x_2$ subject to $x_1+x_2\leq 4$ Does the point $(2,2)$ satisfy the FONC for a local minimizer? The gradient of the objective function is $\nabla f = ...
-1
votes
1answer
16 views

is domain D simply connected? [on hold]

how can we know whether or not domain D is simply connected, by just looking to domain where vector field F is defined?
1
vote
2answers
17 views

Div$f$ is invariant under an orthogonal change of coordinates

Let $f: \mathbb{R^n} \to \mathbb{R^n}$ and $Df$ exists. I need to show that div$f$ is invariant under an orthogonal change of coordinates. Let $T:\mathbb{R^n} \to \mathbb{R^n}$ be an orthogonal ...
2
votes
1answer
32 views

Critical point but not an extremum or saddle point

Let $f: R^2\to R$. Now, a critical point does not mean $f$ has a local (or global) extrema. Of course it could be a saddle point. Does anyone have an example of a function $f: R^2\to R$ that has a ...
0
votes
2answers
65 views

Compute the integral over the volume of a torus,

In $\mathbb R^3$, let $C$ be the circle in the $xy$-plane with radius $2$ and the origin as the center, i.e., $$C= \Big\{ \big(x,y,z\big) \in \mathbb R^3 \mid x^2+y^2=4, \ z=0\Big\}.$$ Let $\Omega$ ...
2
votes
1answer
34 views

Total derivative of $f(A,B)$ , where $f:M(n,\mathbb{R}) \times M(n,\mathbb{R}) \to M(n,\mathbb{R})$

Find the Total derivative of i)$f(A,B)=A+B$ , ii)$g(A,B)=AB$ iii)$h(A,B)=A^2$ where $f,g:M(n,\mathbb{R}) \times M(n,\mathbb{R}) \to M(n,\mathbb{R})$ and $h:M(n,\mathbb{R}) \to M(n,\mathbb{R})$ ...
1
vote
1answer
18 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
votes
0answers
21 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 ...
2
votes
1answer
34 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 ...
2
votes
1answer
55 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 + ...
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 ...
1
vote
2answers
50 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
2answers
47 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 $E \to F\to G\to H$ ...
0
votes
3answers
85 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
vote
4answers
29 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
50 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 $$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1$$ ...
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 ...
1
vote
0answers
24 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
votes
1answer
50 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 ...
0
votes
0answers
63 views

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

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
0answers
49 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 ...
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
vote
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 ...
-8
votes
0answers
41 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.
-2
votes
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 ...
2
votes
0answers
20 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
vote
0answers
21 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
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 ...