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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2answers
43 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
40 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
60 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
89 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
43 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
30 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
26 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
34 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
41 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
24 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
1answer
25 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
24 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
15 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
17 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 ...
2
votes
1answer
38 views

Double integral - Convert to polar coordinates and find the integration limits by a given domain [on hold]

I need help converting to polar coordinates and find the limits of the integrals by this given domain: $$\iint_{D}{} f(x,y)\, dx\, dy$$ $$D= \left\{ (x,y) \mid \dfrac {x^2}{a} \leq y\leq a, -a\leq 0 ...
1
vote
2answers
40 views

If $g:R^m \to R^n$ has derivative $\lambda$ at $a$, is the limit as $x\to 0$ of $\frac{|g(a + x) - g(a)|}{|x|}$ always $\|\lambda\|$?

I am able to show the limit is bounded above by $\|\lambda\|$ if it exists: $$\frac{|g(a +x) - g(a)|}{|x|} \leq \frac{|g(a+ x) - g(a) - \lambda x|}{|x|} + \|\lambda\|$$
1
vote
1answer
32 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 ...
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 ...
0
votes
0answers
26 views

Green's theorem: what does closed path integral mean?

I'm studying green's theorem and having brief idea about this theorem. but little bit confused with first example http://tutorial.math.lamar.edu/Classes/CalcIII/GreensTheorem.aspx (goes to example ...
1
vote
3answers
51 views

Calculating $\lim_{(x,y) \to (0,4)} \frac{xy-4x}{y^2-16}$ or proving it does not exist

I've managed to get the limit into the following form: $y=mx \rightarrow \lim_{x \to 0} \frac{mx^2-4x}{mx^2-16} \rightarrow \lim_{x \to 0} \frac{x (m-4)}{x^2 (m+4) (m-4)}$ I'm not sure how I'm ...
-4
votes
0answers
17 views

the limits of integration for the following iterated integral [on hold]

https://webwork-tr.ncc.metu.edu.tr/wwtmp/math120//gif/e210283-3264-setSummer2015Set3prob4image1.png i want to know how to solve this , i just care about the way to solve it
4
votes
1answer
146 views

Does positive definite Hessian imply the Jacobian is injective?

Suppose $f(x):\mathbb{R}^n \mapsto \mathbb{R}$ is an infinitely differentiable function. If $\nabla^2 f(x)$, the hessian of $f$ is positive definite everywhere, does this imply that the gradient(first ...
1
vote
0answers
39 views

Maximize this function

Let $f(x,y)=x(y\ln y-y)-y\ln x.$ Find $\max_{1/2\le x\le 2}(\min_{1/3\le y\le1}f(x,y))$. This problem is quite easy and it is from Spivak; it is the part $c)$ of the general exercise 2-41 page 43 ...
0
votes
2answers
41 views

Calculating integral using Stokes theorem and directly

Here is my task: Calculate directly and using Stokes theorem $\int_C y^2 dx+x \, dy+z \, dz$, if $C$ is intersection line of surfaces $x^2+y^2=x+y$ and $2(x^2+y^2)=z$, orientated in positive ...
0
votes
1answer
66 views

How to differntiate $\int_{0}^{2\pi} u(re^{i\theta}) d\theta$?

Suppose $u$ is a twice continuously differentiable function on $a< |z|<b, \ z\in \mathbb C,$ which is harmonic that is, it satisfies $u_{rr}+\frac{1}{r}u_r + u_{\theta \theta}=0.$ (If we put ...
0
votes
1answer
21 views

From Euler to Poisson equation: calculus identities suggestions

Kind of things that I always find in books and I never remember. I am looking for a simpler/expanded form of the LHS of the following poisson equation in the case $ \nabla \cdot \underline v =0$ $$ ...
0
votes
0answers
15 views

Momentum method for a gradient descent

I have a reasonable understanding of what a gradient descent is and can apply it properly. Recently while reading about neural networks I have seen in a couple of places that they use a modification ...
1
vote
1answer
48 views

Triple Integral Using Cylindrical Coordinates

Find the total mass of the solid defined by the inequalities $x^2 + y^2 + z^2 \ge 1, \hspace{.1cm} x \ge 0,\hspace{.1cm} y \ge 0$ with mass density $z^2$. I know I have to use triple integrals to ...
5
votes
2answers
45 views

Why does the Hessian work?

I am working through Susskind's 'The Theoretical Minimum' (on physics) – it also includes some maths. In particular, there is an interlude for which he discusses partial differentiation. He discusses ...
1
vote
3answers
45 views

How to evaluate this partial derivative?

If we have some function: $$ f(x,y)= \begin{cases} \frac{xy(x^2-y^2)}{x^2+y^2} &\text {if }(x,y) \neq (0,0) \\ 0 & \text{otherwise}\end{cases}$$ then find $$\frac{d^2f}{dxdy}(0,0).$$ This ...
3
votes
0answers
19 views

Evaluating line integral technique.

I'm trying to do a few questions set by a lecturer on line integrals. I was struggling with a few of them and decided to look at the solutions: Often when $$\int_C \vec{F}\cdot d\vec{r}$$ is tricky ...
-1
votes
0answers
43 views

Surface integral - area and volume [on hold]

Here is my task: Ball S with radius R and center in coordinate beginning is given. Let $\sum $ be its boundary. a) Calculate area of sphere $\sum $ and volume of ball S, which are located between ...
3
votes
3answers
102 views

Is this inequality true? If yes, for what functions?

Let $B=B(0,1)\subset \mathbb R^2$. Let $u$ be a radially symmetric differentiable function on $B$ and $v=Ax+b$ be a linear function where $A$ is a $2\times 2$ matrix satisfies $A=-A^T$, and ...
0
votes
3answers
41 views

How to intuitively arrive at the total derivative limit and the jacobian matrix?

I'm following this PDF and I need to understand how to arrive at the definition of total derivative geometrically. For now, what I understand is that, from the original definition of derivative: ...
2
votes
1answer
28 views

Bound on function increment

consider the function $$ f(x,y) = \dfrac{x^{1/3}}{(x+ay^3)^{1/3}} $$ where $a>0$ is a constant and $x,y\geq 0$. It is easy to see that, outside of the origin, $0\leq f\leq 1$. $f$ itself is not ...
1
vote
0answers
30 views

How do I solve a under-determined quadratic multi-variate system?

I have the following equation: $$ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \beta_{11} X_{1}^2 + \beta_{22} X_{2}^2 + \beta_{33} X_{3}^2 + \beta_{12} X_{1} X_{2} + \beta_{23} X_{2} ...
1
vote
1answer
44 views

Line integrals - Surface area

Here is my task: Calculate surface area of $2(x^{2}+y^{2})^{2}=xy$ between surface $x^{2}+y^{2}=z$ and $z=0$. Here is my attempt to solve this problem. Firstly, I transformed line ...
1
vote
1answer
71 views

Integrating an Iterated Integral

Given the iterated integral: $$\int^{\sqrt{2}}_{-\sqrt{2}}\int^{\sqrt{2-x^2}}_{-\sqrt{2-x^2}}\int^{\sqrt{4-x^2-y^2}}_{\sqrt{x^2+y^2}}{(x^2+y^2+z^2)^{3/2}}dzdydx$$ Now, my question is, what are the ...
-1
votes
1answer
66 views

finding total mass of a solid by inequalities [on hold]

I need help finding total mass of a solid. So this solid is defined by inequalities: $$x^2+y^2+z^2\le 1, x\ge 0, y\ge 0$$ and has a mass density of $z^2$.
0
votes
0answers
10 views

Axysimmetric Poisson equation solution

I'm struggling to find the solution to the follow Poisson problem in spherical coordinates: $ \Delta f(r,\theta) = \sum_{l=1}^{4}k_{l} \left(r\right) P_{l}^{1}\left( \theta \right)$ In the above ...
0
votes
1answer
53 views

Find Total Mass of a Solid

I need help finding total mass of a solid. So this solid is defined by inequalities: $$x^2+y^2+z^2\le 1, x\ge 0, y\ge 0$$ and has a mass density of $z^2$. Maybe we can convert this to cylindrical. ...
3
votes
1answer
42 views

Solving an Iterated Integral

Given the iterated integral: $$\int^{\sqrt{2}}_{-\sqrt{2}}\int^{\sqrt{2-x^2}}_{-\sqrt{2-x^2}}\int^{\sqrt{4-x^2-y^2}}_{\sqrt{x^2+y^2}}{\left(x^2+y^2+z^2\right)^{3/2}}dzdydx$$ Now, my question is, ...
1
vote
2answers
39 views

How I can solve: $\lim_{(x,y) \rightarrow (0,0)} a\cos\left(\frac{x}{\sqrt{x^2+y^2}}\right)$

$$\lim_{(x,y) \rightarrow (0,0)} a\cos\left(\frac{x}{\sqrt{x^2+y^2}}\right)$$ Hi, could someone explain to me how I can solve that limits, i thought that i could studying $L_2=\lim_{(x,y) \rightarrow ...