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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5
votes
1answer
105 views

Continuity ( Functions of 2 variables ).

Given , $$ f(x,y) = \begin{cases} \dfrac{xy^{3}}{x^{2}+y^{6}} & (x,y)\neq(0,0) \\ 0 & (x,y)=(0,0) \\ \end{cases} $$ We need to check whether the function is continuous at $(0,...
2
votes
1answer
59 views

Non linear system of differential equations

Is there a specific name to the following type of non linear ODEs $\begin{array}{c} \dot{x}_1 &= c_1 \, x_2\, x_3 \\ \dot{x}_2 &= \, c_2 x_1 x_3 \\ \dot{x}_3 &= c_3 \, x_2 x_1 \...
4
votes
2answers
189 views

Minimum of an apparently harmless function of two variables

I would like to prove that the minimum of the function $$ f(x,y):=\frac{(1-\cos(\pi x))(1-\cos (\pi y))\sqrt{x^2+y^2}}{x^2 y^2 \sqrt{(1-\cos(\pi x))(2+\cos(\pi y))+(2+\cos(\pi x))(1-\cos(\pi y))}} $$ ...
2
votes
1answer
89 views

Finding the net outward flux of a sphere

Use the Divergence Theorem to compute the net outward flux of: $$ F = \langle x^2, y^2, z^2 \rangle $$ $S$ is the sphere: $$ \{(x,y,z): x^2 + y^2 + z^2 = 25\} $$ First, I took: $$ \nabla \cdot F ...
0
votes
1answer
54 views

How do you go about solving partial differential equations for finding critical points in general optimization problems?

I was reading about partial second derivative test for optimization problems and I came across the example here. I saw the equations have yielded four critical points, but I wasn't able to find those ...
0
votes
0answers
75 views

Partial derivative of recursive exponential $f(x) = \sum^{K_2}_{k_2=1}c_{k_2} \exp(-z^{(2)}_{k2})$ with respect to the deepest parameter

I was trying to take the derivative of the following equation (which can be depicted nicely in a tree like structure, look at the end of question for diagram): $$f(x) = f([x_1, ..., x_{N_p}])= \sum^{...
0
votes
2answers
39 views

Is the funtion $f(x,y)=\frac {x^2y^2}{x^2y^2 + (y-x)^2}$ when $(x,y)\neq (0,0)$ and $f((0,0))=0$ continuous at $(0,0)$ and is this differentiable?

Is the function $$f(x,y)=\begin{cases}\frac {x^2y^2}{x^2y^2 + (y-x)^2} & \text{ , when } (x,y)\not=(0,0)\\0&\text{ , when }(x,y)=(0,0)\end{cases}$$ continuous at $(0,0)$ is this ...
0
votes
1answer
41 views

Conditions on a linear system of ODEs

Let $x:[0,T]\to\mathbb{R}^n$ and $y:[0,T]\to\mathbb{R}^n$ be solutions to an $n\times n$ system of linear ODEs. That is, $$\frac{dx}{dt}=A(t)x+b(t) \mbox{ and } \frac{dy}{dt}=A(t)y+b(t) \mbox{ for } 0&...
1
vote
1answer
67 views

Finding $\iint_S {z \:ds}$ for some $S$

$$\iint_S {z \:ds}$$ In this double integral above, $S$ is the part of a sphere, $x^2+y^2+z^2=1$, which lies above the cone, $z=\sqrt{x^2+y^2}$. How can I calculate the above double integral. Can ...
3
votes
1answer
79 views

Using Stokes' Theorem Finding $\int_C{F\bullet dr}$

Suppose that $C$ is the intersection of $z=2x+5y$ and $x^2+y^2=1$ which is oriented counterclockwise when viewed from above. Now let $$F=\langle \sin{x}+y, \sin{y}+z, \sin{z}+x \rangle$$ How can I ...
0
votes
1answer
65 views

Proper definition use in Stoke's theorem

Let the curve C be a piecewise smooth and simple closed curve enclosing a region, D. Some sources asserts Stoke's theorem to be: $$\oint_{C} F.dr = \iint_{R}\nabla \times FdS$$ Whereas, some claims ...
1
vote
3answers
140 views

How to sketch a surface in a three-dimensional space?

I was asked to hand sketch the surface defined by $$x^2+y^2-z^2=1$$ How could I do that? I find it particularly hard to draw graph in three-dimension, could you give me some advice?
0
votes
1answer
35 views

Finding the derivative of a multivariable function

Suppose $f: \mathbb{R}^n \to \mathbb{R}$ is a differentiable function. Then we can write the derivative of $f$ as a $1 \times n$ row matrix of partial derivatives of $f$ ,i,e, $$Df=\begin{bmatrix}\...
1
vote
2answers
70 views

how to differentiate $y(x) =exp(ax)$ twice

I'm quite confused with this differentiation: Suppose $x$ is a $m \times 1$ column vector, $a$ is a $1 \times m$ vector, I want to differentiate $\exp(ax)$ a few times. I think the first derivative ...
2
votes
2answers
55 views

does simply connectedness require connectedness?

My question consists of two parts. $1)$ suppose domain $D=\{(x,y)\in\mathbb R^2~|~xy>0\}$ is given. Now that is first quadrant and third quadrant with exclusion of $x$ and $y$ axis. We can easily ...
1
vote
1answer
25 views

$F = \langle yz-2xy^2, axz-2x^2y+z, xy+y \rangle$ [duplicate]

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? How can we find an $f$ with $\nabla f=...
0
votes
0answers
83 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
vote
4answers
81 views

Length of Spiral in a plane [closed]

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 $0\leq t<\infty$. ...
0
votes
2answers
299 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? $$\begin{...
2
votes
5answers
122 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
63 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 $M$? ...
1
vote
1answer
50 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 topic,...
1
vote
1answer
44 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
36 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
31 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 $...
1
vote
3answers
144 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} xe^...
1
vote
1answer
40 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
47 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
43 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
56 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
74 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 ...
3
votes
2answers
120 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
49 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
101 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 ...
2
votes
1answer
91 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 = \begin{...
2
votes
2answers
29 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
49 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
99 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
40 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
24 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: $$P=U_{x}=xy^2$$ $$Q=U_{y}=x^2y$$ $$\Longrightarrow U=\int P dx=\frac{x^2}{2}y+C(y)$$ $$ U_{y}=\frac{x^2}{2}+C'(y)=Q=x^2y$$ $$\Longrightarrow \...
0
votes
0answers
124 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 f=u-...
2
votes
1answer
42 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 $\...
3
votes
1answer
99 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
22 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
128 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 ...
-1
votes
1answer
52 views

How to find triple integral of the following question?

$$\int_{0}^{2}\int_{6}^{1}\int_{0}^{1} xyz\sqrt{2-y^2-x^2}\mathbb dx\,\mathbb dy\,\mathbb dz$$ I've been trying to solve that question and over and over again, I get answer of: $$-\left(\dfrac49\...
0
votes
1answer
30 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
60 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 $E \to F\to G\to H$ $$E=(0,0)\,,F=(\pi,0)\;,G=(\pi,\frac{\pi}{...
0
votes
3answers
112 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 \...
2
votes
4answers
82 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) ...