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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0answers
23 views

Proving that $\frac {\partial X} {\partial u} = \frac {\partial (F,G)/\partial (y,u)}{\partial(F,G)/\partial (x,y)}$

The two equations $F(x,y,u,v)=0, G(x,y,u,v)=0$ determine $x$ and $y$ implicitly as the functions of $u$ and $v$, say $x = X(u,v)$ and $y = Y(uv)$. Show that $$\dfrac {\partial X} {\partial u} = ...
0
votes
2answers
59 views

Calculus partial derivatives problem; how can I prove this relationship?

Can anyone help me with this proof, I've attached my working out so far: Show that: $\phi =Ae^{-\frac{kt}{2}}sin(pt)cos(qx)$ satisfies the equation $$\frac{\partial \phi^{2}}{\partial ...
0
votes
1answer
74 views

Calculating $E[x]$ from even probability density function $f_{XY}$

I'm a new user on this site, and I have a question about calculus and probability. I want to prove that $E[x] = \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} xf_{XY}(x; y)dxdy = 0$ when $E[x]$ is ...
2
votes
0answers
91 views

Intersection between a sphere and a plane and parametrization

I am given that a circle is formed when the unit sphere $x^2+y^2+z^2=1$ intersects the plane $x+y+z=0$. I would like to find the equation of that circle using cylindrical coordinates so that I later ...
0
votes
1answer
77 views

trigonometric parametrization

I am trying to figure out a pattern. I will start with examples. $$\text{Let } PD(\text{Set } A):= \text{Parametric Description of }A$$ $$ A:=\{(x,y)\in \mathbb R ^2|x^2+y^2 =1 \} $$ $$PD(A): ...
1
vote
1answer
219 views

proof that a vector-valued function is lipschitz continuous on a closed rectangle

This is a text in my book: The problem is that I have never seen a mean value theorem for vector-valued functions. I have seen it for functions of several varaibles, but then it must be a real ...
0
votes
1answer
66 views

Determine if the derivative $f'(0;a)$ exists for every vector $a$ and compute it's value in terms of the components of $a$.

Let $f(x,y) = \dfrac {xy^3}{x^3 + y^6 }$ if $(x,y) \ne (0,0)$ and define $f(0,0)=0$ Determine if the derivative $f'(0;a)$ exists for every vector $a$ and compute it's value in terms of the components ...
1
vote
1answer
76 views

Is $f(x,y)$ differentiable at the origin?

Let $f(x,y) = \dfrac {xy^3}{x^3 + y^6 }$ if $(x,y) \ne (0,0)$ and define $f(0,0)=0$ Determine if $f$ is differentiable at the origin or not. Attempt: $D_1f(0,0) = \lim_{h\rightarrow 0} \dfrac ...
0
votes
2answers
28 views

If $g = (~g_1(x),g_2(x),…,g_n(x)~)$ is a vector valued function, $f$ is a scalar field, $h=f[g(x)] $ , then what is the value of $\nabla h(x)?$

If $g = (~g_1(x),g_2(x),...,g_n(x)~)$ is a vector valued function, $f$ is a scalar field, $h=f[g(x)] $ , then what is the value of $\nabla h(x)?$ Attempt: $h(x) = f[g(x)] \implies h(x) = (~ ...
2
votes
0answers
34 views

Computing $\int_C (xz+1) \,\mathrm{d}x + (yz + 2x) \,\mathrm{d}y$ with Stokes's theorem - verifying my calculation

I'm not sure if I'm doing this right. I'll write out what I've done so far and if anyone could point out any mistakes, I would really appreciate it. Let $C$ be the curve of intersection of $y + z = ...
6
votes
1answer
152 views

$ \int_1^2\int_1^2 \int_1^2 \int_1^2 \frac{x_1+x_2+x_3-x_4}{x_1+x_2+x_3+x_4}dx_1dx_2dx_3dx_4 $

Evaluate $$I= \int_1^2\int_1^2 \int_1^2 \int_1^2 \frac{x_1+x_2+x_3-x_4}{x_1+x_2+x_3+x_4}dx_1dx_2dx_3dx_4$$ Answer Options: $1$ $\frac{1}{2}$ $\frac{1}{3}$ $\frac{1}{4}$ I need some ...
0
votes
2answers
89 views

Finding a parameter such that two curves intersect orthogonally

For which value of $p$ do the curves $2x^{2} + 3y^{2} =1$ and $px^{2} + 6y^{2} =1$ intersect orthogonally? (Options:) $1/3$, $4$, $3$, $2/3$. I have calculated $\text{grad } f$ for these two ...
0
votes
1answer
37 views

if $h(x) = f[g(x)]$, then prove that $\nabla h(a) = \sum_{k=1}^n D_k f(b)~ \nabla g_k(a)$

If $h(x)=f[g(x)],$ where $g=(g_1,\cdots, g_n)$ is a vector field differentiable at $a$ and $ f$ is a scalar field differentiable at $b=g(a)$. Then prove that $$\nabla h(a) = \sum_{k=1}^n D_k f(b)~ ...
2
votes
2answers
42 views

Problem with proof of multivariable limit

I've got a problem with this limit: ${\lim_{(x,y) \to (0,0)} \frac{ x^{5} + 2y^{3} }{ x^{4} + y^{2} }}$ Can you help me, please? I think it equals 0, but I don't know how to prove it.
0
votes
1answer
34 views

Integration by Parts - Relativistic Energy

I was trying to derive the kinetic energy for a relativistic particle, and I came across this article http://en.wikipedia.org/wiki/Kinetic_energy#Relativistic_kinetic_energy_of_rigid_bodies. I was ...
1
vote
2answers
405 views

Partial Derivatives and the Fundamental Theorem of Calculus

I am being asked to evaluate the 1st-order partial derivatives $-$ $f_{x}$(x,y) and $f_{y}(x,y)$ $-$ of the following multi-variable function: $f(x,y) = \int_{y}^{x} cos(-1t^2 + 3t -1) dt$. Any help ...
4
votes
1answer
212 views

Winding number integral/index in plane

Let $B(x_0,y_0)$ be an open unit disk. Assume that $F(x,y)=(f_1 (x,y),f_2 (x,y)):\overline{B(x_0,y_0)}\rightarrow \mathbb{R}^2$ is a diffeomorphism. Assume also that $F(x,y) \ne (s_0,t_0) $, for all ...
0
votes
2answers
107 views

Least squares minimization of point distances (nonlinear)

We have two sets of 2D points $\bar{x}\leftrightarrow \bar{x}'$ (the bar denotes a vector, i.e. $\bar{x}=(x,y)^{T}$). I would like to minimize discrepancy between the points using the least squares ...
3
votes
0answers
55 views

Function of class C(1) except at a point

Let $f:\Omega\to\mathbb R$, $\Omega\subseteq\mathbb R^n$, a continuous function and of class $C^{(1)}(\Omega\setminus \{ x_0\})$, $x_0\in\Omega$. Suppose that the limits $$l_i=lim_{x\to ...
2
votes
1answer
79 views

Calculation of the normal derivative on a sphere

Fix $y\in B(0,R)$ (open ball) and define for $x\in B(0,R)\setminus \{y\}$ with $$ N(x;y):=\begin{cases} \dfrac{-1}{2\pi}\log\dfrac{rr^*\rho}{R^3},& y\not=0\\ ...
0
votes
1answer
69 views

Describing the shape of a level surface given functions

(1) Describe the level surfaces of $f(x,y,z) = sin(2x+y-z)$. For what values of 'c' do level surfaces exist? For this one I set the function equal to c and tried to put it in a more manageable form. ...
2
votes
1answer
77 views

What are the steps to this functional derivative problem?

I'm trying to derive equations from Matthew Beal's Thesis, Variational Algorithms for Approximate Bayesian Inference pg.55, but I'm stuck on one of the equations (well I'm stuck on a lot of equations ...
0
votes
1answer
17 views

Does type of a critical point depend on domain?

Suppose we have a function $f: D \to \mathbb{R}$, for a domain $D$, and $X \in D$ is a saddle point of $f$. Is it possible that if we constrain $f$ to a new domain $C$, where $C \subset D$, the same ...
0
votes
1answer
86 views

Chain rule and vector function

I am trying to numerically evaluate partial derivatives of the following expression: $$p(\mathbb{x})=(L\circ R)(\mathbb{x})=L(R\mathbb{x})$$ where $$R_{a}=\begin{pmatrix}a & 1 \\ 0 & ...
0
votes
1answer
45 views

Directional derivative of $f(x,y,z) = x^2+y^2-z^2$ at $(3,4,5)$ along the curve of intersection of $2x^2+2y^2-z^2=25$ and $x^2+y^2=z^2$.

Directional derivative of $f(x,y,z) = x^2+y^2-z^2$ at $(3,4,5)$ along the curve of intersection of the two surfaces $2x^2+2y^2-z^2=25$ and $x^2+y^2=z^2$. Attempt: The curve of intersection of ...
1
vote
1answer
33 views

Finding arc length parametrization

Find an arc length parameterization for $r(t)= (e^t\sin(t), e^t\cos(t), 8e^t)$. Tried to follow normal steps and got $t=\ln(s/\sqrt{66})$. Not sure where I went wrong.
0
votes
1answer
293 views

Finding Arc Length Parametrization

Find an arc length parameterization of the line $y=6x+7$. Confused on how to start with x's and y's.
0
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2answers
262 views

What is the operator “capital D” and how can the chain rule be used in this way

I ought to know this but I was somehow always able to avoid mixing partials and $d$s My book notes the following: (with some intermediate steps which are less important than the time taken to write) ...
0
votes
3answers
41 views

Quick question on multivariate differentiation

Does $$\frac{\partial ^2f}{\partial x \partial y}$$ mean diff it wrt $x$ THEN wrt $y$? Or diff wrt $y$ then wrt $x$?
0
votes
2answers
47 views

Multivariable: How can I show the limit does not exist?

How can I show the limit does not exist? $$\lim_{(x,y)\to(0,0)}\frac{xy^2}{(x^2+y^4)\sqrt{x^2+y^2}} $$ I'd rather not using polar representation.
4
votes
1answer
156 views

Integration over the intersection of the $n$-ball and a hyperplane

Let the $n$-ball of radius $r$, centred at $\mathbf{x}_0$, which will be denoted as the region $$ U = \{\mathbf{x}\in\Bbb{R}^n\colon\|\mathbf{x}-\mathbf{x}_0\|^2\leq r^2\}, $$ and is shown in the ...
0
votes
0answers
228 views

Implicit function where the Jacobian determinant is zero

When we have an implicit function defined by $f(x,y)=0$ where Jacobian determinant of $\frac{\partial f}{\partial x}$ is zero Let $x \in \mathbb{R}^n$, $p \in \mathbb{R}$ and $\phi:\mathbb{R}^n \to ...
0
votes
1answer
44 views

Find the extremals of a functional of the form $\int^{x_1}_{x_0}F(y',z')dx$

I was working on Problem 3 in Ch. 2 of Gelfand & Fomin's Calculus of Variations, which reads: Find the extremals of a functional of the form $$\int^{x_1}_{x_0}F(y',z')dx$$ given that ...
0
votes
2answers
31 views

problem with Line integral of vector field

Taking the xyz-coordinate system with $i,j,k$ are the unit vector of each axis, there is a Vector Field $F = {5x+y, 3y-2xz, z} = (5x+y)i + (3y-2xz)j + zk$ I want to find the integral of F on the line ...
1
vote
2answers
32 views

Proving that a function $g:\Bbb{R}^2\rightarrow \Bbb{R}, (x_1, x_2)\mapsto g(x)=\frac{x_1x_2}{x_1^2+x_2^4}$ is not continuous at $x=0$

I would like to prove that the following function $g:\Bbb{R}^2\rightarrow \Bbb{R}$ is not continuous at $x=0$ $$ g(x)=\frac{x_1x_2}{x_1^2+x_2^4} $$ if $x\ne 0$ and equal to $0$ if $x=0$. It's pretty ...
0
votes
1answer
324 views

Find the surface area of the portion of the cone $z^2=x^2+y^2$ that is inside the cylinder $z^2=2y$.

(1) I have solved the problem, but I am not sure about the number of octants the surface covers (this affects the final answer value). (2) Also, I have questions regarding the intersection of the ...
0
votes
1answer
614 views

Volume of Solid Region Between Sphere and Paraboloid

"Find the volume of the solid region above the sphere $x^2+y^2+z^2 = 6$ and below by the paraboloid $z = 4-x^2-y^2$" I am, of course, going to be solving this double integral by converting to polar ...
2
votes
1answer
95 views

Clarification on unit normal vector to a graph of a function

In solving for the unit normal vector $\hat{n}$ of a surface $z=h(x,y)$, the unit normal vector is defined as follows: $$\hat{n}=\frac{\nabla f}{|\nabla f|}$$ with $f=f(x,y,z)=z-h(x,y)$ and $|\nabla ...
1
vote
1answer
55 views

Surface/Path Integral Approach - Brain Fart?

Many times I have dealt with path and surface integrals of the following form $$\int_C \mathbf{F}\cdot d\mathbf{r} \,\,\,\,\,\textrm{(path integral)}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\int_S ...
2
votes
1answer
45 views

Showing the Existence of Total Derivatives

I was presented with the following problem regarding a function that has discontinuous partial derivatives: $$ f(x,y) =\begin{array}{lr} x y \sin(\frac{1}{x^2 + y^2}) : (x,y) \neq 0\\ 0 : (x,y) = ...
1
vote
3answers
78 views

Parametrization of the intersection of two given surfaces

Find a parametrization of the intersection between the two curves $z=x^2-y^2$ and $z=x^2+xy-1$. I figure I should set them equal to each other but I'm not sure where to go from there: $$x^2-y^2 = ...
0
votes
1answer
19 views

If $f ' (x ; y)=0$ for every $x$ in open convex set, then $f$ is constant on open convex set.

$f′(x;y)=0$ for every $x$ in an open convex set $S$ and every $y$ in $\mathbb{R}^n$, Prove that $f$ is constant on $S$. $f′(x;y)$ is the derivative at $x$ in the direction $y$. Seems like I have ...
1
vote
1answer
136 views

Lie Derivative of a section on a vector bundle

I'm still trying to figure out how to do the Lie derivative of a Jacobian. (c.f. earlier unanswered post). If I know how how to do Lie derivatives on section of vector bundles, that would be ...
3
votes
1answer
84 views

Proving that the iterated limit and the two dimensional limit are same

If $\lim_{(x,y)\rightarrow (a,b)} f(x,y) = L$ and if the one dimensional limits : $\lim_{x \rightarrow a}f(x,y)$ and $\lim_{y \rightarrow b}f(x,y)$ both exist, prove that : $$\lim_{x \rightarrow a} ...
3
votes
1answer
22 views

Differentiating a multivariable function

Knowing that $$z(x,y)=f(\frac{x}{y})$$I'm supposed to find $$x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y}$$ . This problem makes no sense to me, can anyone help with the ...
2
votes
2answers
239 views

Limit of a function with two variables: when do we stop looking for another value?

So for instance we have this limit of a function $\displaystyle\lim_{(x,y)\to(0,0)}{xy\over \sqrt{2x^2+y^2}}$, and the function isn't continuous at the point $(0,0)$. Now we can try to find the limit, ...
1
vote
1answer
59 views

Understanding directional derivatives

I am confusing myself when it comes to directional derivatives and gradients. The gradient of a function shows the direction of the greatest change. So when we chose a unit vector as the direction to ...
0
votes
0answers
17 views

Which of the following vector fields can be a gradient of a function?

Which of the following vector fields can be a gradient of a function? $A. F(x, y) = (2x + y, x + 2y)$ $B. F(x, y) = (2xy + y, x + 2y^2)$ $C. F(x, y) = (2x + y, x + 2)$ D$. F(x, y) = ...
1
vote
0answers
20 views

Is a mapping $\mathbb{R}^2 \to \mathbb{R}^2$, with harmonic components and positive Jacobian, injective?

Let $F = \left( \begin{array}{c} f \\ g \end{array} \right) \colon \mathbb{R}^2 \to \mathbb{R}^2$ be a mapping, each of its two components are harmonic functions in the plane, i.e. $\Delta(f) \equiv ...
0
votes
2answers
74 views

How can I parametize a curve of intersection of two surfaces?

To find out directional derivative $f(x.y.z)=x^2+y^2−z^2$ at $(3,4,5)$ along the curve of intersection of the two surfaces $2x^2+2y^2−z^2=25$ and $x^2+y^2=z^2$ I am trying to parametrize above two ...