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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4
votes
1answer
34 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
58 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
36 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
21 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
36 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 ...
3
votes
2answers
30 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
57 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
25 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
64 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
78 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
62 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 ...
2
votes
1answer
68 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
24 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
40 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
79 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
41 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
32 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
27 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
113 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
29 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: ...
2
votes
0answers
36 views

Why are these two things equivalent when doing surface integrals?

As I understand it, when doing a surface integral we have, $$\iint_S F\cdot ndS=\iint_D r~\frac{r_a \times r_b}{|r_a \times r_b|}|r_a \times r_b|dA$$ and this is true because $$ndS=\frac{r_a \times ...
1
vote
2answers
30 views

How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$?

How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$? I want to parametrise so I can use the divergence theorem to calculate the flux along the surface above. I don't know how to do it and would like ...
0
votes
1answer
43 views

Integration with respect to dx, dy and dz (More than one variable)

Sorry if my title was vague but i was not entirely sure what its called. Anyways i was solving some work and energy problems and encountered this integration: $$\int_{2,1,4}^{2,-3,3} 2x\sin^2y ...
1
vote
2answers
25 views

How to calculate a surface integral using Gauss' Divergence theorem.

I'm trying to evaluate the following: $$\iint_S F\cdot n~dS$$ given $S$ is defined to be the surface area of the cylinder given by $$x^2+y^2 \leq 1, 0 \leq z \leq 1$$ and ...
0
votes
2answers
34 views

Chain rule for implicit functions

Let $F_1(x_1,x_2,x_3)=f(x_1,f(x_1,x_2,x_3),x_3)$ and $F_2(x_1,x_2,x_3)=f(x_1,x_2, f(x_1,x_2,x_3))$. Find $\displaystyle \frac{\partial F_i}{\partial x_j}$ for all $i=1,2$ and $j=1,2,3$. I know ...
0
votes
1answer
22 views

How to calculate $\iint_S~F \cdot n dS$ for the following.

How to calculate $$\iint_S~F \cdot n dS$$ when $n$ is the unit normal vector to the surface, $F(x,y,z)=(x,y,z)$ and the surface in question is $$x^2 - y^2 + z^2 = 0,~ y \in [0, 1] $$ So far here is ...
0
votes
1answer
23 views

Is there a method to parameterise any surface? And how could I parametrise this one given?

I'm having major trouble every time I need to parametrise a surface in order to take a surface integral, I just have no idea where to even start half of the time. Is there some kind of method that can ...
2
votes
1answer
25 views

Using Green's theorem to find an area.

I wish to find out the area enclosed by the ellipse $C:=2x^2+3y^2=2y$ using Green's theorem. I know how to parametrize the ellipse and understand Green's theorem I just don't understand how it is ...
0
votes
1answer
16 views

Signs of a point of intersection between a paraboloid and tangent plane

So I've calculated the value in the subject line but I get signs opposite to the professor. The original question is find the point on the paraboloid $$z = 4x^2 + y^2$$ at which the tangent ...
0
votes
0answers
18 views

Incorrect Signs on Tangent Planes

So basically I've calculated a tangent plane to a surface, and a normal line for that plane through the point where the surface touches the plane, and I'm getting signs opposite to the professor's ...
1
vote
1answer
16 views

Prove that $f(v_1, v_2)$ is greater 0 $\forall v_1, v_2$

I have the function $f_{a, b, c}\colon \mathbb{R}^2 \to \mathbb{R}$, $f_{a, b, c}(v_1, v_2) = av_1^2 + 2bv_1v_2 + cv_2^2$. I want to know for which $a$, $b$ and $c$ this function $f_{a, b, c}(v_1, ...
1
vote
1answer
22 views

Functions with rank $n$.

An open set $U\subset \mathbb{R}^n$ contains the closed origin-centered unit ball $B=B(0,1)$. If a $C^1$ mapping $f:U\rightarrow \mathbb{R}^n$ with rank $n$ obeys $\|f(x)-x\|<1/2$ for all $x\in ...
0
votes
1answer
43 views

Expressing $\cos(\varphi x)$ as a function of $x\sin\varphi,x\cos\varphi$

Let $\varphi,x\in\mathbb{R}$. I wonder if one can explicitly express $\cos(\varphi x)$ as a function of the variables $x\sin\varphi$ and $x\cos\varphi$. Suppose we denote ...