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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15
votes
4answers
407 views

How do I generalize the derivatives / integrals from multivariable calc?

$\newcommand{\RR}{\mathbb{R}}$ This is a long post (EDIT, two years later: not as long as my answer), so I'll put the big question right at the top: There's a whole lot of derivative-like and ...
1
vote
1answer
34 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
0answers
39 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
2answers
16 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.
0
votes
1answer
34 views

Findng 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 ...
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
36 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 ...
1
vote
4answers
78 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: ...
3
votes
3answers
77 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
36 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
24 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
0answers
90 views

Proving $\sigma$-additivity and interchanging order of summation/integration just because positive

Let $\Omega = {\omega_1, \omega_2, ...}$ be some countable set. Let $\mathfrak{F} = 2^{\Omega}$. Consider a sequence {$p_n$} in [0,1] s.t. $\sum_{n=1}^{\infty} p_n = 1$. Define P: $\mathfrak{F} \to ...
1
vote
1answer
7 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
37 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
15 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, ...
2
votes
1answer
304 views

Vector Calculus (Gradients, Potential Functions, and Equipotential Curves)

This is my first question on the Mathematics section of StackExchange, so please forgive me if I don't follow all the rules or things like that. Here's my question: Consider the following potential ...
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
53 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 ...
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 ...
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.
2
votes
1answer
171 views

Spivak Calculus on Manifolds, Theorem 5-2

In the proof Theorem 5-2 of Spivak Calculus on Mannifolds how is \begin{align*} V_2\cap M=\{f(a):(a,0)\in V_1\}? \end{align*} (That $\{f(a):(a,0)\in V_1\}=\{g(a,0):(a,0)\in V_1\}$ is clear.) Edit: ...
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 ...
6
votes
4answers
112 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?
0
votes
2answers
294 views

taylor expansion in cylindrical coordinates

If I have a function Y($r$,$\theta$) in cylindrical polar coordinate system, then how do I Taylor expand this function around some point ($r_0$,$\theta_0$)? I want the exact formula for Taylor ...
3
votes
3answers
141 views

Evaluating $\int_0^{\infty} \frac{\sin x}{x} dx$ with Fubini theorem.

I have to calculate $\int_0^{\infty} \frac{\sin x}{x} dx$ using Fubini theorem. I tried to find some integrals with property that $\int_x^{\infty} F(t) dt = \frac{1}{x}$, but I cant find anything else ...
-4
votes
0answers
51 views
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 ...
0
votes
2answers
60 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, ...
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 ...
0
votes
1answer
57 views

Why does Continuous Partial Differentiability Imply Total Differentiability?

Let $f: \mathbb{R}^d \to \mathbb{R}$ be such that the partial derivatives $\frac{\partial f}{\partial x_i}:\mathbb{R}^d \to \mathbb{R}$ exist everywhere and are continuous. Then show that $f$ is ...
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 ...
1
vote
0answers
35 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 + ...
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: ...
7
votes
1answer
2k views

Singular jacobian matrix?

I have a series of questions, in various degrees of befuddled muddledness (and they are related to my previous questions: this and this) First question: how do I do a change of variable if the ...
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 ...
1
vote
5answers
76 views

Optimize over unit circle to prove $|ax + by| \le \sqrt{a^2 + b^2}$

I have the following problem which, straight off the shelf, seems totally approachable. It's been giving me difficulty however: Let $a,b,x,y \in \mathbb{R}$, and suppose that $x^2 + y^2 =1$. ...
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 ...
44
votes
6answers
6k views

Intuitive interpretation of the Laplacian

Just as the gradient is "the direction of steepest ascent", and the divergence is "amount of stuff created at a point", is there a nice interpretation of the Laplacian (a.k.a. divergence of gradient)? ...
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
42 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
1answer
33 views

How to solve this vector equation for optical flow

I am unable to solve for $\textbf{h}$ in the following equation $\sum\limits_{\textbf{x}=1}^n2\partial F(\textbf{x})/\partial\textbf{x}(F(\textbf{x}) + \textbf{h}^{T}\partial F(\textbf{x})/\partial ...
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 ...
1
vote
3answers
114 views

Confounding Lagrange multiplier problem

Optimize $f(x,y,z) = 4x^2 + 3y^2 + 5z^2$ over $g(x,y,z) = xy + 2yz + 3xz = 6$ According to the theorem the gradients must be parallell, $\nabla f = \lambda \nabla g$, so their cross product must ...
0
votes
1answer
101 views

Firm non-expansiveness in the context of proximal operators

$\newcommand{\prox}{\operatorname{prox}}$ Probably the most remarkable property of the proximal operator is the fixed point property: The point $x^*$ minimizes $f$ if and only if $x^* = \prox_f(x^*) ...
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 ...
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
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
0answers
28 views

Finding a Gradient Vector given only a derivative and direction

I've seen this question answered but in $2$ variables (which means you have two equations and two unknowns and can solve simultaneously), but not this variation, and I'm stuck. So given that the ...
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, ...