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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0
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1answer
328 views

Find the volume under the surface

Find the volume under the surface $z=f(x,y)$ over the rectangle R, where $f(x,y)=x^4+xy+y^3$ and $R=[1,2]\times[0,2]$
-4
votes
2answers
131 views

Find the center of mass with a region r [closed]

Can someone help me with this? I don't know what to do for this problem at all. Thanks
0
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2answers
47 views

Graphing on a region with integral

Can someone help me with this I am very lost in this
3
votes
2answers
121 views

Prove that $\frac{1}{2\pi}\frac{xdy-ydx}{x^2+y^2}$ is closed

I would like to prove that $\alpha = \frac{1}{2\pi} \frac{xdy-ydx}{x^2+y^2}$ is a closed differential form on $\mathbb{R}^2-\{0\}$ . However when I apply the external derivative to this expression ...
1
vote
1answer
75 views

Solution of Lagrangian

Could you give me advice how to solve the following Lagrangian? $$L=x^3+y^3 - \lambda (x^2-xy+y^2-5)$$ $$ \left\{ \begin{array}{c} \frac{\partial L}{\partial x} = 3x^2 - \lambda (2x-y) = 0 ...
1
vote
4answers
340 views

Triangle integral with vertices

Evaluate $$I=\iint\limits_R \sin \left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)\, dA,$$ where $R$ is the triangle with vertices $(0,0),(2,0)$ and $(1,1)$. Hint: use ...
3
votes
1answer
135 views

Laplacian on Sphere of Function Only Depending on Angle Between Points

Consider a function $f:S^2 \to \mathbb{R}$ , with $S^2$ the unit $2$-sphere in $\mathbb{R}^3$. Let's say that $f$ depends only on the polar angle $\theta$ from the north pole (e.g., $f(r,\theta,\phi) ...
0
votes
0answers
124 views

SN Evaluate of a intergal

Use SN EVALUATE to evaluate $$\int_0^1\int_0^{\sqrt{1-x^2}}e^{-(x^2+y^2)}\,\text{d}x\,\text{d}y.$$ I do not understand what SN means can you please help me?
2
votes
1answer
321 views

Triple Integral over a shifted sphere

I am interested in the following: Let $f(x,y,z)$ be a given (known) function in Cartesian coordinates, and let $B_d(p_0) = \{ y: ||y-p_0|| < d \}$ (i.e., a sphere centered at p_0). I want to ...
0
votes
2answers
79 views

How to set up a triple intergal with $x, y,$ and $z$

Use a triple integral to find the volume of the solid bounded by $z=16xy$, $z\ge 0$, $0 \le x \le 5$, $0 \le y \le 4$. I know how to set up the integral for $x$ and $y$ it would be $0$ to $5$ for $x$ ...
0
votes
1answer
110 views

How to set up a double integral with $x,y$ and $z$?

Use a double integral to find the volume of the solid bounded by graphs of the equations given by: $z=xy^3, Z>0,\; X>0,\; 5X<Y<5$ How would you set up this integral? please help me.
3
votes
1answer
83 views

$F$ is incompressible $\iff$ $G$ is incompressible

If $F, G$ are vector fields, and $F(G(x,y)), G(F(x,y)): \mathbb{R}^2 \rightarrow \mathbb{R}^2 \rightarrow \mathbb{R}^2$ are the identity $(x,y) \mapsto (x,y)$, prove that F is incompressible $\iff$ G ...
2
votes
2answers
108 views

Travel distance of a particle

How can we show that a projectile fired at an angle $\theta$ with initial speed $v_0$ travels a total distance $\frac{v_0^2}{g}\sin2\theta$ before hitting the ground? The way I set it up is: direction ...
1
vote
2answers
185 views

How to find global extremum when constraint isn't compact set

Sometimes, the constraint is not a compact set. As a result, the local minimum may not be global. For example, $ f=x^2+y^3$ subject to constraint $ x+y=4/3$. Using Lagrange multiplier method, I ...
2
votes
2answers
336 views

Evaluate the double integral using substitution:

Evaluate: $$\int_0^1\int_0^\sqrt{1-x^2}e^{-(x^2+y^2)}\,\mathrm dy\;\mathrm dx$$ I am trying to use substitution on this problem by making $x^2+y^2 = u$ and $\sqrt{1-x^2} = v$. I then tried to add ...
3
votes
1answer
54 views

Integral of the gradient of a semilinear function.

Let $u:\mathbb{R}^n \rightarrow \mathbb{R}$ be a semilinear map so that for any $k\in \mathbb{R}$ the surface $\partial C_k^+=\{x:k=u(x)\}$ is contained in the union of finitely many hyperplanes. ...
0
votes
1answer
85 views

Integral to find the area with bounded regions.

Use an iterated integral to find the area of the region bounded by $$4x - 7y = 0$$ $$x + y = 8$$ $$y = 0$$ I need help in starting this problem. Can someone help me start it as in give me a idea ...
1
vote
0answers
40 views

Integrate the exponential of sum of circular differences?

Given positive integer $N$ and parameters $T>0$, $a$, $b$, what is $\int_{t_1=0}^T \cdots \int_{t_N=0}^T e^{a(t_1+\cdots+t_N)+b(|t_1-t_2|+\cdots+|t_{i-1}-t_i|+|t_N-t_1|)} dt_1 \cdots dt_N$ ? Any ...
2
votes
1answer
95 views

Vector-by-Vector derivative

Could someone please help me out with this derivative? $$ \frac{d}{dx}(xx^T) $$ with both $x$ being vector. Thanks EDIT: I should clarify that the actual state I am taking the derivative is $$ ...
0
votes
1answer
101 views

if f(x,y) = 1 for all (x,y) where R has a nice shape like a rectangle or triangle what is another geometric interpretation of…?

If $f(x,y)$ is greater than or equal to $0$ on region $R$ in the plane, then $\iint_R f(x,y)dA$ can be interpreted geometrically as the volume of the solid under the surface $z=f(x,y)$ and above $R$. ...
2
votes
1answer
213 views

Derive the curl in general coordinates.

Many mathematical physics texts inform the orthogonal curvilinar coordinates system and differential operators. But, they don't use the precise mathematical method for the derivation of the curl and ...
0
votes
1answer
30 views

As for the sufficient conditions for the domain $\Omega$ in Green's Theorem

The wiki page says that if a domain is enclosed by a simple, closed, piecewise, parametrized curve, then Green's Theorem can be applied on it. But the textbook, Advanced Calculus by Fitzpatrick, says ...
1
vote
0answers
155 views

Find the equation of the tangent plane to the level surface

(a) Find the equation of the tangent plane to the level surface $V (x, y, z) = V (3, 4, 5)$ at the point whose $x$ and $y$-coordinates are $3$ and $4$, respecively, where $$V (x, y, z) = ...
3
votes
1answer
523 views

Find the volume inside both $x^2+y^2+z^2=4$ and $x^2+y^2=1$.

What is the volume inside both $x^2+y^2+z^2=4$ and $x^2+y^2=1$? The chapter I am working on is called Change of Variables in Multiple Integrals, for my Vector Calculus class. I understand that we ...
6
votes
3answers
253 views

Lagrange multiplier method, find maximum of $e^{-x}\cdot (x^2-3)\cdot (y^2-3)$ on a circle

I attempted to design an exercise for my engineer students and couldn't solve it myself. Maybe here are some experts in calculus who have some better tricks than I do: The exercise would be to ...
1
vote
1answer
117 views

What is the exact, rigorous, full statement of Divergence (Gauss') Theorem in $\mathbb{R}^3$ (without being too complicated)?

The wolfram page http://mathworld.wolfram.com/DivergenceTheorem.html states the formula $$ \int_{V} \nabla \cdot \mathbf{F} dS = \int_{\partial V} \mathbf{F} \cdot d\mathbf{S} $$ but it does not speak ...
1
vote
2answers
261 views

Targets of Fighter plane

If fighter plane travels along the path $$r(t)=(t-t^3,12-t^2,3-t),$$ how can we show that the pilot cannot hit any target on the x-axis? Any pointers and hints would be appreciated. I do not ...
1
vote
1answer
92 views

A differential form to compute the k-volume of a k-parallelogram in n dimensions

Computing the k-volume of a k-parallelogram (i.e. a parallelogram spanned by k n-dimensional vectors) in n dimensions is straightforward: Let $P=[\overrightarrow{v_1},...,\overrightarrow{v_k}]$, then ...
2
votes
2answers
158 views

Gauss Theorem example

Verify the Gauss theorem for the vector field $F(x)=\frac{x }{\|x\|},$ where $x \in W \subset \mathbb{R}^3$ and $$W=\left\{(x,y,z) \in \mathbb{R}^3 \left/ a^2\right.\leqslant x^2 + y^2 + z^2 \leqslant ...
5
votes
2answers
121 views

Solving a 5 dimensional function in a neighbourhood

Consider a function $f:\mathbb{R}^5 \to \mathbb{R}^2$ defined by $$f(u,v,w,x,y)=(uy+vx+w+x^2,uvw+x+y+1)$$ such that $f(2,1,0,-1,0)=(0,0)$ (i) Show that we can solve $f(u,v,w,x,y) = (0,0)$ for ...
1
vote
2answers
124 views

$\mathbf{F} = \nabla f$ Get the function f, when given a vector field

For conservative vector fields F the following equation holds $$\int_c \mathbf{F} \cdot d\mathbf{r} = \int_c \nabla f \cdot d\mathbf{r} = f(\mathbf{r}(b)) - f(\mathbf{r}(a))$$ Now I have a lot of ...
2
votes
1answer
43 views

Finding multiple integral on bounded area.

Today I just learn on multiple integral. Somehow this question quite confusing. Find the area of the 1st quadrant region bounded by the curves y=$x^3$, y=2$x^3$ and x=$y^3$, x=4$y^3$ using ...
3
votes
2answers
62 views

Multiple integrals with $x_1 < x_2< \dots < x_n$

I was looking at an example with the following integral: $$\iiiint_{0 \le x \le y \le z \le t,\ 0 \le t \le \frac{1}{2}} 1 \,dx\,dy\,dz\,dt = \frac{1}{16}$$ Is it true in general that $$\int \dots ...
1
vote
1answer
60 views

Jacobbian Transformation Multiple Integral

The question says : Sketch the region under the transformation of u=x+y and v=y for $$R=\{(x,y): 0\leqslant x\leqslant 1 , 0\leqslant y \leqslant1\}$$ Find the Area of the region. Given answer is ...
2
votes
2answers
75 views

Line integal of vector field depends only on the shape and orientation of the curve?

My question is about line integral of vector field in multi-variable calculus. As you know well, the line integral of a vector field over some parametrized curve $X(t)$ is independent of ...
1
vote
1answer
65 views

Parametric representation of $\sqrt{x^2+y^2}\le z \le 2$

just wondering how to parametrize this. Question is: Let $C$ denote the conical region $\sqrt{x^2+y^2}\le z \le 2$. Find a parametric representation $\mathbf{x}(u,v)$ for $S$, the surface of $C$. ...
3
votes
1answer
613 views

Irrotational Vortices

When trying to find some further information regarding irrotational flows, I encountered the notion of an "irrotational vortex": http://en.wikipedia.org/wiki/Vortex#Irrotational_vortices In the ...
1
vote
1answer
124 views

the implication of zero mixed partial derivatives for multivariate function's minimization

Suppose $f(\textbf x)=f(x_1,x_2) $ has mixed partial derivatives $f''_{12}=f''_{21}=0$, so can I say: there exist $f_1(x_1)$ and $f_2(x_2)$ such that $\min_{\textbf x} f(\textbf x)\equiv ...
3
votes
3answers
325 views

Prove an analog of Rolle's theorem for several variables

On p. 135 of Buck's Advanced Calculus, he asks the reader to prove an analog of Rolle's theorem for functions of two variables (I suspect the number two is arbitrary). The hint is to assume that $f=0$ ...
2
votes
1answer
104 views

Integral on surface

Please help me calculate integral on surface. $$\iint\!(2-y)\,dS $$ $$ 0\leq z\leq 1;\, y =1 $$ I can't understand what should I do with '$z$' coordinate. I assume I should do something with it, not ...
2
votes
2answers
33 views

What is the definition of curl of $\mathbf{F}(x_1, x_2) = ( F_1 (x_1,x_2) F_2(x_1,x_2))$ in $\mathbb{R}^2?$

What is the definition of curl of $\mathbf{F}(x_1, x_2) = ( F_1 (x_1,x_2) F_2(x_1,x_2))$ in $\mathbb{R}^2?$ Most textbook says only of vector fields in the space $\mathbb{R}^3$...
4
votes
2answers
355 views

Calculating area of astroid $x^{2/3}+y^{2/3}=a^{2/3}$ for $a>0$ using Green's theorem

question as follows. Show that for any planar region $\Omega$, $$\mathrm{area}\left(\Omega\right)=\frac{1}{2}\oint_{\partial\Omega}(xdy-ydx).$$ Use this result to find the area enclosed by the ...
0
votes
2answers
65 views

Bounds of $\oint_{\partial R}\left((x^2-2xy)dx+(x^2y+3)dy\right)=\iint_{R}\left(2xy+2x\right)dxdy$

I'm just having some trouble figuring out the bounds and boundary of the following integral. Question as follows: Evaluate $$\oint(x^2-2xy)dx+(x^2y+3)dy$$ around the boundary of the region contained ...
3
votes
1answer
2k views

Distance of a test point from the center of an ellipsoid

I'm trying to learn about Mahanalobis distance and I'm pretty close to getting the idea. I've learned that the distance has got a lot to do with the properties of an ellipsoid. I have understood so ...
0
votes
1answer
197 views

equation of the tangent plane

Find an equation of the tangent plane to the surface at the given point. $g(x, y) = x^2-y^2$ at $(7, 2, 45)$. I know the answer is between $14(x-7)-4(y-2)+(z-45)=0$ or $14(x-7)-4(y-2)-(z-45)=0$. I ...
2
votes
2answers
133 views

extrema and saddle points

Examine the following function for relative extrema and saddle points: $$f(x, y) = 9x^2-5y^2-54x-40y+4.$$ I did this and got that the point should be at $(3, -4, 3)$. Is that right? Also, how do I ...
1
vote
3answers
58 views

Gradient of a function in multi-variate calculus please help

Find the gradient of the function at the given point. $g(x, y) = 3xe^{y/x}$, at point $(3, 0)$. How do you compute the gradient of this function. Please help me.
0
votes
1answer
31 views

Computing directional derivative of $f(x,y,z)$

Find the direction derivative of $f(x,y,z) = xy + yz + xz$ at the point $P(1,1,1)$ in the direction of $v = \langle7,3,-6\rangle$. I got the answer as $32/\sqrt{94}$. Is that right? If not what is the ...
1
vote
1answer
247 views

Find the equation of the tangent plane given a vector instead of point

Find the equation of the tangent plane at $\mathbf p = (0,0)$ on the surface $z=f(x,y)=\sqrt{1-x^2-y^2}$. Give an intuitive geometric argument to support the result. However $\mathbf p$ is a ...
1
vote
1answer
651 views

How to examine if multivariable functions are differentiable?

How to examine if functions: $f(x,y)=|x+y|$ and $g(x,y)=\sqrt{|xy|}$ are diffirentiable in points: $(0,0)$ for $f(x,y)$ and $(0,1)$ for $g(x,y)$