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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0answers
40 views

Classify the degenerate point;

Given $$f(x,y)=y^4+y(x-1)^2-8y^2$$ Find the three critical points, use Hessian method to classify the two non degenerate points. Then ; By considering the value of $f$ along the curve ...
2
votes
2answers
72 views

Show this function has infinitely many critical points and classify;

Show that $$f(x,y)=x^2-x+\cos(xy)$$has inifinitely many critical points and classify; Partial Derivative w.r.t. $x$ $$f_{x}=2x-1-y\sin(xy)=0$$ Partial Derivative w.r.t. $y$ ...
1
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1answer
26 views

2nd derivative test fail

I trying to solve this problem in Advanced Calc by Buck, sec 3.6 problem 9: Let $f(x,y)=(y-x^{2})(y-2x^{2})$. Show that the origin is a critical point for $f$ which is a saddle point, although on ...
5
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0answers
35 views

Find the volume of the region bounded by the planes $ z=8-y^2, y = 8-x^2, x=0, y=0, z=0$

I figured out the bounds for z: $z=0$ to $z=8-y^2$ The bounds for y: $y=0$ to $y=8-x^2$ The bounds for x: $x=0$ to $x=\sqrt{8}$ (Since $8-x^2 = 0$) So, the volume by using triple integral: ...
0
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0answers
15 views

Multiple Integral Approximation

Does anybody know how to simplify following multiple integration ? ...
0
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2answers
26 views

Inconclusive Second derivative test ,Now how shall i proceed

While doing maxima and minima questions i have encountered upon question in which i cannot show nature of points P1 : Given $f(x,y) = 2x^4 - 3x^{2}y + y^{2}$ Doubtful case is as origin . P2: ...
0
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1answer
29 views

Geometric Meaning of Parital Derivative (specific problem)

I have an article in my handouts which I doubt is wrong. Article is following: "Geometric Meaning of Partial Derivatives Suppose $z = f ( x , y )$ is a function of two variables.The graph of $f$ is a ...
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1answer
17 views

Proof with curve smoothness.

I've got 2 problems. Let $f:\mathbb{R} \rightarrow \mathbb{R}^2$ $:$ $Im f = f(\mathbb{R}) = \{|x|+|y|=1\}$. Show that $ \exists_{t\in\mathbb{R}}$ : $f'(t)=(0,0)$ 2.$f:\mathbb{R} \rightarrow ...
1
vote
1answer
64 views

Shape of level curves and Hessian determinant (Calculus 3)

Consider the surface $$Q(x,y)=Ax^2+Bxy+Cy^2$$ and the Hessian determinant $4AC-B^2$. How does the sign of the Hessian determinant determine the shape of $Q$'s level curves? When are the ...
0
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1answer
27 views

Line integral $\int_C(3x + \sin(y)) ds$ where $C$ is line segment from $(1,2)$ to $(5,4)$

How to calculate the line integral $$ \int_C(3x + \sin(y)) \,ds $$ where $C$ is line segment from $(1,2)$ to $(5,4)$? Progress I know $r(t) = \langle 1+4t,2+2t\rangle$ from $(0,1)$ so I have an ...
1
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1answer
32 views

maximize function with two variables

I would like to maximize the function: $f(x,y) = c[x\log(2y) + (1-x)\log(2(1-y))]$ subject to constraint that $x,y \in (0,1)$ to find a relationship between $x$ and $y$ that maximizes $f(x,y)$. My ...
4
votes
1answer
23 views

Remainder in the multivariate Taylor expansion

For the function $f:\mathbb{R}\to\mathbb{R}$, I can write the Taylor expansion $$f(x+h) = f(x) + f'(x)h + \frac{1}{2!}f''(x)h^2 + O(h^3)$$ where the remainder is $o(h^2)$ as well. I'm confused with ...
0
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0answers
25 views

Using Feynman's Subscript Notation

I have a homework problem that wants me to calculate the force $\vec{F} = \vec{\nabla}_{\vec{X}}U + \frac{\mathrm{d}}{\mathrm{d} t} \left(\vec{\nabla}_{\dot{X}} U\right)$ where $U(\vec{X}, \dot{X}, ...
1
vote
5answers
115 views

Domain, range and zeros of $f(x,y)=\frac{\sqrt{4x-x^2-y^2}}{x^2y^2-4xy^2+3y^2}$

Given the following function with two variables: \begin{equation} \frac{\sqrt{4x-x^2-y^2}}{x^2y^2-4xy^2+3y^2} \end{equation} I need to find a) the domain for the above function. Can anyone give me ...
0
votes
2answers
46 views

Double integral: Stokes' theorem

Honestly, I have no idea what Stokes' theorem is. I only know that the circulation can be found by this theorem. Would anyone mind helping me?
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0answers
17 views

Dimensions of rectangular parallelopiped of maximum volume…

i Need to calculate volume of parallelopiped of maximum volume with edges parallel to the coordinate axes that can be incribed in a ellipseoid $(x/a)^{2} + (y/b)^{2} + (z/c)^{2} =1$ . Apparently ...
0
votes
2answers
25 views

What is an example for the mean-value theorem to fail to hold for differentiable functions $\mathbb{R}^{n} \to \mathbb{R}^{m}$ if $m \geq 2$?

We know that for any integer $n \geq 1$ and $m=1$, the mean-value theorem holds for differentiable functions $\mathbb{R}^{n} \to \mathbb{R}^{m}$. Nevertheless, what is an example for the mean-value ...
2
votes
1answer
60 views

What is the purpose of the symbol $\int f(x)dx$? [closed]

In elementary calculus I was taught that given $x\in\mathbb{R}$ (or $\vec x\in\mathbb{R}^n$) if the limit $$\lim_{\delta\to 0}\sum_{i=1}^n f(x_i)\Delta_i\ \ \ \ (\ \text{or }\lim_{\delta\to ...
3
votes
2answers
37 views

Lagrange Multiplier Question and my attempt

Question is Find the extrema of $xyz$ when $x+y+z=a$ , a>0. Strating with usual Lagrange Multiplier method i get $f_x$ = $yz$ +$\lambda$ =0 $f_y$ = $xz$ +$\lambda$ ...
4
votes
1answer
23 views

Notation Question with Line Integrals over Vector Field

Previous Question: What is the Convention in Arc Length Parametrization? This is a follow-up question to my previous post on line integrals going in opposite directions. At first I thought I ...
2
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1answer
30 views

Finding Triangle with constant perimeter and largest area (Lagrange Multiplier)

Question is to find Finding Triangle with constant perimeter and largest area by method of lagrange multiplier . What i have done is that i have firstly taken $x+y+z=2k$ , where x,y,z are sides of ...
1
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2answers
40 views

Find Rectangle of Constant Perimeter whose diagonal is maximum (My attempt with Lagrange Multipliers)

Question is to Find Rectangle of Constant Perimeter whose diagonal is maximum (My attempt with Lagrange Multipliers) . I took rectangle with sides $x$ and $y$ . Since Perimeter is constant so i ...
2
votes
1answer
25 views

Volume between a cone and and an Hyperboloid

I'm trying to use integration in several variables to find put what is the volume between the cone $x^2+y^2=z^2$ and the hyperboloid $x^2+y^2=3+z^2$ I'm having a hard time with this problem, as the ...
0
votes
1answer
24 views

Find flow of function over polygon

The problem: Evaluate $\displaystyle\int_{C} (x^{4}+5y^{2})\,dx + (xy-y^{5})\,dy$, where C is the polygonal path joining the vertices $\left[\begin{array}{c}-1\\0\\\end{array}\right], ...
1
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0answers
18 views

Limit of 2 variables function with a parameter

Find for which $\alpha \in \Re$ the function is continuous in $(0,0)$ $$ f(x,y) = \begin{cases} \frac{-2x^3 arctan(y)}{(x^2+y^2)^\alpha}, & \text{(x,y) $\neq$ (0,0) } \\ 0 & \text{(x,y) ...
1
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0answers
51 views

Taylor's Theorem for Multivariable Implict Functions

I'm trying to find the $2$nd order Taylor polynomial for $z=g(x,y)$ near the point $(\frac {\pi}{2}, 1,1)$, given the function $\sin(xyz)=z^2$. I've never found the Taylor polynomial of a function ...
3
votes
1answer
28 views

Change of Variables: which order of integration limits?

An exercise in Larson/Edwards Calculus (10th ed.) asks the reader to evaluate the double integral $$\int_R\int 4(x^2+y^2)\,\mathrm{d}A$$ using a given change of variables ...
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0answers
6 views

Difference between Relative And Absolute Extrema and concept of convex regions

Can anybody explain to me difference between relative and absolute extrema ? Secondly somebody have told me on stackexchange that for convex regions(triangles,rectangles) extrema occurs on corners ...
2
votes
2answers
54 views

Can Anyone help me with Lagrange multiplier problem

I need to find absolute maximum and minimum of thi function $$F(x,y) = x^{2} - y^{2} - 2y$$ over $$R = \{ (x,y)\ |\ x^{2} + {y^2} \leq 1\} $$ Thanks for help
2
votes
1answer
63 views

Evaluate the surface integral from the paraboloid

Evaluate the surface integral $$\iint\limits_S xy \sqrt{x^2+y^2+1}\,\mathrm d\sigma,$$ where $S$ is the surface cut from the paraboloid $2z=x^2+y^2$ by the plane $z=1$. Is it possible for the ...
4
votes
4answers
310 views

Multivariable limit

I'm studying multivariable limits and I have a problem regarding this limit: $$\lim_{(x,y)\to(0,0)} (x^2+y^2)^{x^2y^2}$$ . I've found that it is equal to 1, by rewriting the limit using $t = ...
-1
votes
4answers
60 views

How to solve these two equations $2x + y = 1/x^{2}$ , $ x +2y = 1/y^{2}$ [closed]

How do I solve the following system of two equations, two unknowns? $2x + y = 1/x^{2}$ $ x +2y = 1/y^{2}$
0
votes
0answers
13 views

Finding critical points of function of two variables and my attempt

Given $f = x^{2} + 2bxy + y^{2}$ $f_x = x +by =0$ $f_y = bx+ y=0$ On olving these two equations i get $y( 1 - b^{2})=0$ Thus i have got either $y=0 $ or $b^{2} = 1$ Corresponding to $y=0$ i ...
0
votes
1answer
29 views

Convergence test for improper multiple integral

I have a function $f:\mathbb R^n \to \mathbb R$ such that $f(x)=(1+|x|)^me^{-\frac{|x|^2}{a}}$. I need to check is $$\int\limits_{\mathbb R^n}f(x)dx = \int\limits_{\mathbb R^n} ...
1
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0answers
36 views

Rectangle surmounted by an isosceles triangle

A window has the shape of a rectangle surmounted by an isosceles triangle. Determine the dimensions of the window, if its if its perimeter is to be at most $M$ and its area is to be maximized. I have ...
2
votes
0answers
17 views

Multivariable chain rule with vector valued function

Suppose $f:\mathbb{R}^n \rightarrow \mathbb{R}$, $\mathbf{g}:\mathbb{R}^n \rightarrow \mathbb{R}^n$ and $\mathbf{x} \in \mathbb{R}^n$. How do I find a formula for $\nabla f(\mathbf{g}(\mathbf{x}))$?
1
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1answer
27 views

about $\nabla^{4}\Phi=0$, write down this equation in terms if Cartesian Coordinates

$\nabla^{4}\Phi=0$, write down this equation in terms if Cartesian Coordinates(x,y). I am a bit confused here, the question doesn't tell you if $\Phi$ is scalar or vector, but i think it is a vector, ...
0
votes
0answers
13 views

Find the volume of the solid bounded by 3 equations (using triple integrals)

Find the volume of the solid bounded by z=(x^2)+(2y^2), z=0, and x+y+2z=2. I used the triple integral of dzdxdy, and integrated with respect to z first with the bounds of z as (0 to (1-(x/2)-(y/2))) ...
0
votes
1answer
20 views

Maximum and minimum of function in a curve

Find the points of maximum and minimum of the function $$f(x,y,z) = 2x + y - z^2$$ in the compact space $$C = \{(x,y,z) \in \mathbb{R}^3 : 4x^2 + y^2 -z^2 = -1,z\ge 0, 2z \le 2x + y + 4\}$$ So, I ...
2
votes
3answers
39 views

Limit of two variable does not exist?

I got this limit on my midterm: $$\lim_{(x,y)\to 0,0} \frac{(x^2+2x-4y^2+4y)} {(x+2y)} $$ and if I were to plug in $y=-\frac{x} {2}$ it gives me that the limit equals $\frac{0} {0}$ as opposed to the ...
0
votes
1answer
19 views

Line Integral Given with dy Instead of ds

$\int_C(x^2y^3-\sqrt{x})\,\mathrm{d}y$, $C$ is the arc of the curve $y=\sqrt{x}$ from $(1,1)$ to $(4,2)$ I set $y=t$ and therefore $x=t^2$ on the interval as $t$ goes from $1$ to $2$ and also ...
3
votes
1answer
64 views

$f(b)-f(a) =((b-a)/2)\cdot(f'(a)+f'(b))-((b-a)³/12)\cdot f'''(c)$ [duplicate]

Let $f$ be three times differentiable on $[a,b]$. $f'''$ is continuous. Show that there is a $c\in[a,b]$ such that $$f(b)-f(a) =((b-a)/2)\cdot(f'(a)+f'(b))-((b-a)³/12)\cdot f'''(c)$$ This looks like ...
1
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2answers
28 views

Line Integral Over the Right Half of a Circle

$\int_Cxy^4\,\mathrm{d}s$, $C$ is the right half of the circle $x^2+y^2=16$ I solved for $x$ and got $x=\sqrt{16-y^2}$ but I'm pretty stumped about where I should go. I know I need to get $x$ and $y$ ...
2
votes
3answers
60 views

Find maximum/minimum for $\cos(2x) + \cos(y) + \cos(2x+y) $

I have not been able to find the critical points for $\cos(2x) + \cos(y) + \cos(2x+y) $
-1
votes
5answers
69 views

Find the maximum and minimum value of $x^{4} + y^{4} + z^{4} -4xyz$ [closed]

I am having problem in finding critical points of this $x^{4} + y^{4} + z^{4} -4xyz$
1
vote
1answer
78 views

How to find the area inside $x^2+y^2+\sin(4x)+\sin(4y)=4$ using Green's Theorem

This was a modification of a previous question I asked, except now I'm saying how to solve the area inside $$x^2+y^2+\sin (4x) +\sin (4y) = 4 $$ The equation is a simple closed shape. Here's is the ...
1
vote
0answers
32 views

Lagrange multipliers question and my attempt

Question is to minimise the $f(x,y)$ = $3x^{2} + y^{2} - x $$$$$ and constraint is given by $2x^{2} + y^{2} =1 $ Question is simple and ii have got most of points but i seem to miss few points ...
4
votes
1answer
39 views

How to use Lagrange Multiplier in this question?

I have to find absolute maximum and minimum values of $f(x,y)$ = $4x^{2} + 9y^{2} -8x - 12y + 4 $ over rectangle in first quadrant bounded by lines $x=2 , y=3$ and coordinate axes I have checked ...
1
vote
2answers
52 views

Parameterizing a surface

The question I was asked goes like this: The part of the hyperboloid $5x^2 − 5y^2 − z^2 = 5$ that lies in front of the yz-plane. Let x, y, and z be in terms of u and/or v. Find a parametric ...
0
votes
1answer
25 views

Multivariable Maclaurin Series

Find Maclaurin series for a) $\cos(x + y)$ b) $\frac{\log(1 + x)}{(1 + y)}$. Honestly, I'm really just confused about the process of finding a Taylor series expansion for multivariate equation.