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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2
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2answers
75 views

Which Cross Product for the Desired Orientation of a Hyperboloid ? [Stewart P1103 16.9.8]

P1103 16.9.$8.$ Evaluate the surface integral $\iint_S \mathbf{F} \cdot d\mathbf{S}$. $\mathbf{F} = (x^3y,-x^2y^2,-x^2yz)$ and $S$ is the surface of the solid bounded by the hyperboloid $x^2 + ...
2
votes
0answers
51 views

Homework Stokes' theorem

I'm not seeing what i did wrong here for this vector calculus problem. If anyone could point me in the right direction i would be most appreciative. The problem reads: Let $$ F= (2yz, -x+3y+2,x^2 +z) ...
0
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0answers
24 views

Convergent Improper Integral help

I am currently studying improper integrals and came across the following problem. Analyze the convergence of the improper integral of $f(x,y) = 1 / ( x^4 + y ^2 ) $ over $R = \{(x,y) : x\geq 1, y\geq ...
2
votes
2answers
60 views

basic triple integral

I am pretty confident I can solve this question so please don't give me the answer, but I am having trouble "imagining" the area they are referring to. Question: calculate $$\iiint_D ...
2
votes
0answers
187 views

Proof that Curl and Divergence Uniquely Define Vector Field

Why/how do curl and divergence yield a unique vector field? Can anyone give me a proof? Also can you construct a vector field from any curl/divergence?
1
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1answer
43 views

Locally minimizing a concave function

What will happen if we minimize a concave function via gradient descent? Where does it get stuck? Intuitively a concave function has more structure than an arbitrary function, and seem to be easier ...
1
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3answers
53 views

How do you know when you'e found all the critical points? Is there a rule?

For example, for $f\left(x,y\right)=y^3+3x^2y-6x^2-6y^2+2$, $f_x\left(a,b\right)=$ $6xy-12x$ and $f_y\left(a,b\right)=$ $3y^2+3x^2-12y$ How many critical points am I looking for? How do I ...
1
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1answer
50 views

Line Integrals in $\mathbb{R}^n$ and Differential 1-Forms

I am really lost in the notation of my book and am looking for some conceptual help. So... for a differential 1-form $\omega$, every value of $x\in \Omega$ maps to a linear function $\omega _x$, ...
0
votes
1answer
104 views

A simple double integral - check my answer

Pretty simple question but I have no one to compare with. We are asked to find the integral $\int \int _A (x^2-2y)dA$ where $A$ is the triangle $\{(1,0),(2,1),(3,4)\}$ What I did: I found the ...
1
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1answer
67 views

Parametrisation of an ellipse in polar coordinates

If we have an ellipse with equation $x^2/a^2+y^2/b^2=1$ then if we were to change this into polar coordinates then would the parametrisation be $x=ar\cos(\theta)$ and $y=br\sin(\theta)$? Also what ...
0
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1answer
31 views

Find double integrals boundaries.

I have to solve this integral $$\iint_D xy {\,\rm d}x {\,\rm d}y$$ where the domain $D$ is delimited by: $x^2+y^2=4$ and $y^2=3x$; My problem is that I don't know how should I plot those and more ...
2
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1answer
113 views

The meaning of Differentials in Integration

This is further to the questions discussed in a previous post Here is an example of what I mean: Suppose that $C$ is a closed path in the plane and consider the line integral of $xy\,dx+x^2\,dy$ over ...
2
votes
2answers
59 views

Point on $z = \frac{1}{xy}$ closest to origin

Where $x>0$ and $y>0$. I want to work with the square of the distance formula from the origin, so I went with $f(x,y) = x^2 + y^2 + \frac{1}{(xy)^2}$. Then I found the first partial ...
0
votes
1answer
23 views

Find volume of an object $y=1-x^2$ $\ldots$

The $X$ and $Y$ axis, and curve $y=1-x^2$ define an area $\mathbf{A}$ from the first quarter of an $XY$ plane ($x \text{ and } y \ge 0$). An object is directly above $\mathbf{A}$ between planes $z=1$ ...
1
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0answers
56 views

volume in spherical coordinates

I am trying to find the region bounded by the sphere $p = 2\cos\psi$ and hemisphere $p=1$, $z\geq 0$. Not quiet sure not to do this problem, so please help.
0
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1answer
34 views

Eliminate unknown function $f$ by obtaining a PDE

Question Let the funcion $z = z(x,y)$ be given by the equation $z = xy + f(x^2 -y)$, where $f$ is an arbitrary $C^1$ function. By forming the first partial derivatives of $z = z(x,y)$: $p = z_x$ and ...
0
votes
1answer
33 views

Find if the following limit exists

This might me a simple problem, but I want to see if I got it right Given $$F(x,y) = \left\{\begin{array}{b}\frac{xy}{x^2+y^2}& \text{if }x-3y=0\\ y+3&\text{ if }x-3y\neq 0 ...
-1
votes
3answers
48 views

$f(x,y)= (x^4-y^4)/(x^4+y^4)$ and $ f(0,0)=0 $ show that f is continuous at origin. Using epsilon- delta definition

$f(x,y)= (x^4-y^4)/(x^4+y^4)$ and $f(0,0)=0$ show that f is continuous at origin. Using the epsilon- delta definition. This is all I know till now: we have to prove: $|(x^4-y^4)/(x^4+y^4)| < ...
1
vote
1answer
46 views

Double Integral Help $(x^2+y^2+a^2)^{-2} dx \, dy$

Hi I'm currently revising for a maths module that I am taking as part of my physics degree. All was going well until I hit a dead-end with this integral, any ideas how to evaluate it? $$ ...
0
votes
1answer
30 views

Transforming an ellipse

I am trying to find an integral over the positive quadrant of the ellipse $\frac {x^2}{a^2}+ \frac {y^2}{b^2}= 1$ by transforming the variables $ x= a\sin \theta \cos \phi $ $ y= a\sin \theta ...
1
vote
0answers
15 views

When to use spherical and when to use cylindrical coordinates?

How do we know when to use spherical coordinates and when to use cylindrical coordinates? What do we base it on, the surface or the equation of the function we are given? e.g in a question I'm given ...
0
votes
1answer
21 views

Need help with the formula of the normal to a surface when using Stokes' theorem!

When using stokes theorem how do we find the normal to a surface? What is the formula for the normal if the surface is parametrized and what is the formula when the surface is not parametrized. Also ...
0
votes
1answer
56 views

When and why must we parameterise $f(x, y) = …$ with variables besides $x, y$?

For 10C, my choice of parameterisation $\mathbf{r} (x,y) = ( x, y, z(x, y))$ fails to effect the right answer, but that of user ellya does function. Yet for 9C, the parameterisation $\mathbf{r} (x,y) ...
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vote
2answers
94 views

Intersection curve between a circle and a plane - Stokes theorem

What is the intersection curve between the circle $$x^2+y^2=1$$ and the plane $$x+y+z=0$$ If i am not wrong, I should solve the equation system \begin{align} x^2+y^2-1=0 \\ x+y+z=0 \end{align} But I ...
1
vote
2answers
84 views

reverse the order of integration

How do you reverse the order of this integral into $dy\,dx$? I feel like you need two separate ones but I don't know how to do it: $$\int_0^3\int_\sqrt{y}^3 f(x,y) \, dx \, dy$$ thanks
0
votes
3answers
739 views

Finding the speed of a particle (parametric math)

I have to find the speed (as a function of $t$) of a particle whose position at time $t$ seconds is represtented by $$c(t)=(\sin t+t, \cos t+t)$$ How would I go about finding the maximum speed? ...
1
vote
1answer
80 views

Without Stokes's Theorem - Calculate $\iint_S \operatorname{curl} \mathbf{F} \cdot\; d\mathbf{S}$ for $\mathbf{F} = (-y^3,x^3,z^3)$ - 2012 9C

Question: 2012 9C. Consider the (cutoff) paraboloid defined by $z= x^2 + y^2 , \frac{1}{9} \le z \le 1$. Sketch the surface. Verify Stokes’s Theorem for for $\mathbf{F} = (-y^3,x^3,z^3)$. Herein, I ...
0
votes
1answer
38 views

Find the length of the curve

I need to find the length of the curve $$c(t)=(3e^{t}-3, 4e^{t}+7)$$ for $$0\le t \le 1$$ If I understand correctly, I need to take the derivative of the y part of that coordinate over the ...
0
votes
1answer
86 views

With Stokes's Theorem - Calculate $\iint_S \operatorname{curl} \mathbf{F} \cdot\; d\mathbf{S}$ for $\mathbf{F} = (-y^3,x^3,z^3)$

2012 9C. Consider the (cutoff) paraboloid defined by $z=x^2 + y^2, \frac{1}{9} \le z \le 1$. Sketch the surface. Verify Stokes’s Theorem for for $\mathbf{F} = (-y^3,x^3,z^3)$. Herein, I ask ...
5
votes
2answers
72 views

Find the parametric equation to the curve

Find the parametric equation for the curve. $$x^{2}+y^{2}=10$$ I haven't learned parametric equations fully yet, so I wanted to check with you guys and see if you can confirm if I'm doing this ...
0
votes
1answer
63 views

Determine Cross Product with Left Hand vs Right Hand

If I perceive http://en.wikipedia.org/wiki/Cross_product correctly, then to determine the cross product With a right hand, let: the 1st vector in the cross product = your index finger = in red ...
0
votes
1answer
38 views

Evaluating an integral using the Jacobian help!

For this question I have managed to sketch the region however I don't understand how to solve this sort of a problem where we have two regions? Also when we use the change of variable formula with ...
0
votes
1answer
46 views

Express the parametric equation in form of y=f(x)

I need to express the parametric equation in the form of $y=f(x)$ by eliminating the parameter. I haven't learned how to do this yet, I've attempted to read a few pages though but they didn't help me ...
1
vote
3answers
215 views

Need to extremize the function $f(x,y)=x^2+y^2$.

Determine the points on the curve $$x^4+y^4=1$$ that are closest and furthest away from the origin. Explain why this corresponds to extremizing the function $f(x,y)=x^2+y^2$ under the condition ...
0
votes
1answer
33 views

evaluating a flux integral

Question: "Region V, of unit volume, is bounded by the closed surface S. Given the vector field $\mathbf{F}=\langle 7x,2y,5z\rangle$, evaluate: $$\int_S \mathbf{F}\cdot\mathbf{dS}$$ I guessed that ...
1
vote
1answer
63 views

Solving Simplified Hamilton's Equation

I have a question on a project that I am working on. I have included a large amount of the background information so that all relevant information is included, however the question is as follows (it ...
1
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2answers
109 views

Proof Strategy - Without Stokes's Theorem - Calculate $\iint_S \operatorname{curl} \mathbf{F} \cdot\; d\mathbf{S}$ for $\mathbf{F} = yz^2\mathbf{i}$

2013 10C. Question: Consider the bounded surface S that is the union of $x^2 + y^2 = 4$ for $−2 \le z \le 2$ and $(4 − z)^2 = x^2 + y^2 $ for $2 \le z \le 4.$ Sketch the surface. Use suitable ...
1
vote
2answers
350 views

Inverse of a two variable function?

How can I find the inverse of the following bijective function: $$ F(x,y) = (x-y^2, x + y - y^2) $$ I usually do inverses by using the matrix method, but I have no clue on how to face this problem
0
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1answer
546 views

Triple integral limits over pyramid

I am just studying triple integrals in my calculus class and I have the following problem as homework and I am not sure how to start: Let $P$ be a pyramid defined by the base [-2,2]x[-2,2] in XY plane ...
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2answers
138 views

How can I prove the surjectivity of the following function

$F:\mathbb{R}^2 \to \mathbb{R}^2,\\ f(x,y)= ((x^3)-x),y)$ How can I check if this is surjective or not?
4
votes
1answer
139 views

Product of Elements in SU(2)

Let $$ V := \frac{x_4+i\vec{x}\cdot{\vec{\sigma}}}{\left|x\right|}$$ where $\left(x_1,x_2,x_3,x_4\right)\in\mathbb{R}^4$, $|x|$ is the Euclidean norm, and $\sigma^j$ are the Pauli matrices. Let ...
2
votes
1answer
179 views

Learning Advanced Mathematics

I'm a 12th grade student and I've recently developed a passion for mathematics . Currently my knowledge in this particular area is comprised by : single-variable calculus , trigonometry , geometry , ...
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vote
1answer
517 views

Integration by parts of $\varphi \cdot \operatorname{curl}(u)$

Does anybody happen to know the integration by parts formula for $\iint(\varphi\cdot \operatorname{curl}(u) dV)$, where both $\varphi$ and $u$ are 3D vectors? Is there a good reference for similar ...
0
votes
1answer
112 views

Volume between cone and cylinder with a shift

I am trying to solve the following problem: Compute the volume of the solid bounded by the $ \ z \ = \ 3 \sqrt{x^2 + y^2} \ $ , the plane $ \ z = 0\ $ , and the cylinder $ \ x^2 +(y−1)^2 \ = \ 1 ...
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1answer
70 views

Injective and surjective equations

How can I prove that the following equation is both surjective and injective (thus bijective) $$F(x,y) = (x - 2 y^2 +3, y +2 x - 4)$$ ??
2
votes
1answer
55 views

Double integral definition

I don't understand the definition of double integral. For instance in the functions with single variable the definite integral was defined as Riemannian sum as: $$\lim_{n\to\infty}\sum_{k=1}^{n} ...
2
votes
2answers
782 views

The meaning of $\lambda$ in Lagrange Multipliers

This is related to two previous questions which I asked about the history of Lagrange Multipliers and intuition behind the gradient giving the direction of steepest ascent. I am wondering if the ...
1
vote
1answer
69 views

Maximum and minimum of a multivariable function

I have to find the maximum and minimum of the function $$f(x)=\frac {1}{x^2+y^2}$$ when $$x^2+(y-2)^2 \leq 1$$ What I did was: 1) Find the critical points of the function when $x^2+(y-2)^2 < ...
0
votes
1answer
20 views

Tricky parametrization problem

Find $|r(1)|$ if $|r(0)| = 0$ and $(r \text{dot} r')(t) = 6t^2$ $\forall t$. What is the trick for this one? How do I work backwards?
2
votes
2answers
97 views

Gradient and Swiftest Ascent

I want to understand intuitively why it is that the gradient gives the direction of steepest ascent. (I will consider the case of $f:\mathbb{R}^2\to\mathbb{R}$) The standard proof is to note that the ...