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
votes
1answer
8 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
15 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
30 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
43 views

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

Alright, this one seems silly but I got a nasty answer when trying to break it up using long division. How do you integrate $\left(\dfrac{x+y}{x-y}\right)^4$?
2
votes
0answers
31 views

Green's theorem application

Problem Determine all the circumferences $\mathcal C$ 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 ...
0
votes
2answers
17 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
30 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
38 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
19 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
19 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 ...
2
votes
1answer
51 views

Finding the area under the cycloid $x=t-\sin (t),\;y=1-cos (t)$

I need to find the area under the cycloid $x=t-\sin (t),\;y=1-cos (t)$ above axis and between $x=0,x=2\pi$ using $\underline{\text{Green's theorem}}$ I found in Wikipedia this evaluation: ...
0
votes
1answer
61 views

Evaluate $\oint_{C}xy^2dx+2x^2 dy$

$$\oint_{C}xy^2dx+2x^2y dy$$ triangle:$$C=\{(0,0),(2,2),(2,4)\}$$ My attempt: Using Green's theorem $$\oint_{C}\underbrace{xy^2}_{P}dx+\underbrace{2x^2y}_{Q} dy=\iint\bigg(\frac{\partial ...
-1
votes
1answer
77 views

Line Integral: $\int_C{x^2}\:dy$

How can I calculate $\int_C{x^2}\:dy$ in which $C$ is a line segment from the point $(0,0)$ to $(3,2)$? I am new to line integrals, I am only familiar when given a function and in $ds$. How can I do ...
0
votes
1answer
28 views

Change of variables when integrating over a triangle

I want to calculate the integral $$ \iint_D(x-y)dxdy $$ where D is the triangle made up of the vertices (0,0), (-2,1) and (-1,3). (Graph) My idea was to do this substitution $$ \begin{equation} ...
2
votes
1answer
62 views

Calculate $\int_C{y^2\:ds}$ where $C$ is $x^2+y^2=9$

I need help calculating $\int_C{y^2\:ds}$ where $C$ is $x^2+y^2=9$. So what I first did was convert $C$ into parametric and then I set it up like this: $$\int_0^{2\pi}(3\sin t)^2\sqrt{(-3\sin ...
4
votes
2answers
61 views

Evaluating the line integral $\int_C{F\cdot dr}$ for a particular conservative vector field $F$

So I have this two dimensional vector field: $$F=\langle (1+xy)e^{xy},x^2e^{xy}\rangle$$ How can I tell whether $F$ is conservative or not? And also how do I calculate $\int_C{F\cdot dr}$, where $C$ ...
0
votes
1answer
58 views

Evaluate $\iint dydx$ on the domain $0\leq r\leq1$, ${\pi}/{3}\leq\theta \leq{2\pi}/{3}$ [duplicate]

I need to evaluate $\displaystyle\iint \color{red}{dydx}\;\;\;,\bigg\{\frac{\pi}{3}\leq\theta \leq\frac{2\pi}{3}\bigg\}\;\;\;\;,0\leq r\leq1$ $\color{blue}{\text{without using polar ...
0
votes
1answer
31 views

Writing line integral as 1-form

If $F: \Bbb R^n \rightarrow \Bbb R^n $ is a vector field and $\phi : [a,b] \rightarrow \Bbb R^n$ is a continously differentiable path we defined the integral of $F$ along $\phi$ as $\int_{\phi} F = ...
0
votes
1answer
14 views

Calculating min/max of a multivariate function on a region

This video shows an example of how to find the absolute maxima and minima of the function $f=xy+y^2$ at the region $\{(x,y):|x|\leq1,|y|\leq2\}$. I understand why he set $f_x, f_y$ to $0$, checked ...
0
votes
1answer
27 views

Green theorem application

Suppose that a simple closed curve $C$ in the $xy$ plane, that bounds a convex domain $D$ containing the origin. The curve is specified by $x=f(\varphi), y=g(\varphi)$ where $0\leq \varphi< 2\pi$ ...
2
votes
1answer
71 views

Doesn't $x^3+2y^3+3z^3=0$ give a surface in $R^3$?

In my last exam on Advanced Calculus (following Spivak's Calculus on Manifolds), I couldn't solve the following question. True or false: the set $S$ in $R^3$ given by $x^3+2y^3+3z^3=0$ is a ...
3
votes
2answers
82 views

Stokes Theorem. Where is my mistake?

Use Stoke's Theorem to prove that the following line integral has the indicated value. $$ \int_\mathscr{C} y \,dx +z\,dy+x\,dz = \pi a^2 \sqrt{3}$$ where $\mathscr{C}$ is the intersection curve ...
0
votes
0answers
40 views

Question about the gradient of a function?

I was under the impression that the gradient of a function points in the direction of greatest increase of the function. Okay that is fine but I was also reading that it gives a normal vector at a ...
2
votes
1answer
63 views

Does $\int_cf\:dx$ depend on the parameterization of $C$?

As long as we don't switch the orientation, does $\int_cf\:dx$ depend on the parameterization of $C$ or no? I have a feeling that it does not depend. However, can someone give me a rigorous proof as ...
0
votes
1answer
83 views

Finding the area under the curve $y=3-3\cos(t),x=3t-3\sin(t)$

I need to find the area under the curve $\color{blue}{y=3-3\cos(t),x=3t-3\sin(t)}$ and between $\color{blue}{x=2\pi,x=0\text{, above axis}}$ using $\color{blue}{\text{Green's theorem}}$. My attempt ...
2
votes
2answers
19 views

Change of variable (Fourier Transform related)

Consider a problem below... The solution offered to this particular question (1)a)) simply state the change of variable ksi to by to yield the result, I'm failing miserably to see how.
1
vote
2answers
25 views

Finding the flux of $\iint \vec F\hat n\;ds$

I need to find the flux $\displaystyle\iint \vec F\hat n\;ds$ of the vector feild $\vec F=4x \hat i-2y^2\hat j+z^2 \hat k$ throughe the surface $S=\{(x,y,z):x^2+y^2=4,z=0,z=3\}$ My attempt: (I'm ...
2
votes
1answer
45 views

Why does $\nabla F{(x,y,z)}$ point in the direction of greatest increase of the function, and why is $|\nabla F(x,y,z)|$ its slope?

Why does $$\nabla F{(x,y,z)}$$ point in the direction of greatest increase of the function and why is $$|\nabla F{(x,y,z)}|$$ it's slope (I should actually ask what the slope would mean here as I'm ...
3
votes
3answers
80 views

Evaluate $\iint_{R}(x^2+y^2)dxdy$

$$\iint_{R}(x^2+y^2)dxdy$$ $$0\leq r\leq 2 \,\, ,\frac{\pi}{4}\leq \theta\leq\frac{3\pi}{4}$$ My attempt : Jacobian=r $$=\iint_{R}(x^2+y^2)dxdy$$ $$x:=r\cos \theta \,\,\,,y:=r\cos \theta$$ ...
0
votes
1answer
37 views

Intuition behind surface integrals

While line integrals derive their intuition from , and are analogous to, the concept of Work in physics, what intuition is there to back up the notion of surface integrals? In the texts I've been ...
0
votes
1answer
51 views

How does gradient of a vector point steepest ascent

The derivative of distance function with respect to time give velocity function in single variable calculus. But how does gradient of a multivariable function point steepest ascent? I have been ...
1
vote
1answer
9 views

Let $\vec F(x,y)=(y+xg(x),y^2), \vec F(1,1)=(3,1)$. $\vec F_x \perp \vec F_y$.Find $g$.

Let $\vec F(x,y)=(y+xg(x),y^2), \vec F(1,1)=(3,1)$. $\vec F_x \perp \vec F_y$ Find $g$. Attempt: I look for the partial derivatives, I did so differentiating each coordinate with respect to ...
2
votes
3answers
39 views

Proof: $f(x,y)=\sqrt{4x^2+y^2}$ is continuous at $(0,0)$

Prove $f(x,y)=\sqrt{4x^2+y^2}$ is continuous at $(0,0)$. Attempt I need to find a $\delta(\epsilon)$: $$\forall \epsilon>0\exists \delta>0: 0<\sqrt{x^2+y^2}<δ \implies ...
1
vote
1answer
17 views

Function determining temperature of points along a curve

Let $T=x^2+y^2+z^2$ be the function determining the temperature at the point $(x,y,z)$. Find a function that determines the temperature at the points along the curve $\vec\alpha(t)=(4\cos t, 4 \sin t, ...
1
vote
4answers
88 views

Evaluate $\iint dy\,dx;\frac{\pi}{4}\leq\theta \leq\frac{3\pi}{4};0\leq r\leq2$

I need to evaluate $\displaystyle\iint \color{red}{dydx}\;\;\;,\frac{\pi}{4}\leq\theta \leq\frac{3\pi}{4}\;\;\;\;,0\leq r\leq2$ $\color{blue}{\text{without using polar coordinates}}$. My attempt: ...
0
votes
1answer
33 views

Evaluate $\int_{-2}^{2}\int_{y^2-3}^{5-y^2}dxdy$ [duplicate]

In the black I evaluated the integral and I got 64/3, now I need to evaluate the same integral with $\color{red}{dydx}$ .in the $\color{blue}{\text{blue}}$ color is my attempt, I don't think that my ...
0
votes
1answer
54 views

How to prove the limit exists for function of two variables?

Problem: Evaluate the indicated limit or explain why it does not exist: \begin{align*} \lim_{(x,y) \to (0,0)} \frac{x^2 y^2}{x^2 + y^4} \end{align*} The definition of limit my calculus textbook gives ...
1
vote
1answer
28 views

Area surrounded by a curve

I would need help to calculate the area surrounded by a curve. The curve is given with the following polar coordinates: I know we need need to integrate with respect to r and theta but am stuck ...
2
votes
1answer
50 views

How to find $\int_0^1 \int_x^1 \arctan(\frac{y}{x})dxdy$? [duplicate]

How to find $$\int_0^1 \int_x^1 \arctan \left( \frac{y}{x}\right)~dxdy$$ I am not looking for any full solutions just some small hints to get me started would be great.
0
votes
2answers
61 views

Solve Double Integral Using Change of Variables: $\int^1_0 \int^{y^2}_0 {y\cos(x-y^2)dxdy}$

I am currently learning about Jacobians, and I need help on the following integral: $$\int^1_0 \int^{y^2}_0 {y\cos(x-y^2)dxdy}$$ The first thought that came to my mind was change of variables, ...
-4
votes
0answers
51 views

How to evaluate the integral $\int^{1/2}_0\int^{1-x}_x (x+y)^9(x-y)^9 \, dy \, dx$? [on hold]

How to evaluate the integral $\int^{1/2}_0\int^{1-x}_x (x+y)^9(x-y)^9 \, dy \, dx$?
3
votes
1answer
72 views

Area Enclosed by Ellipse with Function: $(x+y)^2+(x+3y)^2=1$

How can I find the area of the following region which is enclosed by the following curve: $$(x+y)^2+(x+3y)^2=1$$ This is an ellipse, and I graphed it to find that its center is at the origin. Not ...
6
votes
4answers
114 views

Calculate $\int^{1/2}_0\int^{1-x}_x (x+y)^9(x-y)^9 \, dy \, dx$

How can I find the following integral: $$\int^{1/2}_0 \int^{1-x}_x (x+y)^9(x-y)^9 \, dy \, dx $$ My thoughts: Can we possibly convert this to spherical or use change of variables?
1
vote
2answers
46 views

How to evaluate the line integral $\int_C (y-z)\,dx+(z-x)\,dy+(x-y)\,dz$

How to evaluate the line integral $\int_C (y-z)\,dx(z-x)\,dy(x-y)\,dz$. The curve $C$ is the intersection of the cylinder $x^2+y^2=1$ and the plane $x-z=1$. I am really stuck on how to to do this ...
1
vote
0answers
36 views

Finding extrema of function of three variables

So i have to study this function and find out if there are any local or absolute extrema : $ f:\mathbb{R}^3 \rightarrow \mathbb{R} :$ $$ f(x,y,z)=2-\left(z-\sqrt{x^2+y^2}\right)^2 + ...
2
votes
2answers
30 views

Function of several variables which is continuous at single point

Examples of functions on $\mathbb{R}$ which are continuous at a single point are well known. But what about $f:\mathbb{R}^2\to \mathbb{R}$ which is continuous at a single point? I tried to proceed as ...
0
votes
1answer
17 views

How are these two terms in $y$ removed from the triple integral? (Divergence theorem?)

I will post the photo here for convenience sake. I wish to understand why it just says, odd in $y$ and then cancels the $y$ bits and simplifies the integral a whole lot. Here is the scan: ...