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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2answers
41 views

What is the derivative of $\dot{x} = f(x(t))$?

I am supposed to take the derivative of a function similar to this one: Take the derivative of $$\dot{x} = \cos(x)$$ where $x$ is a function of $t.$ I believe that this can be generalized to the ...
0
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0answers
17 views

unit normal vector to paraboloid surface

I have a question about unit normal vectors the the surface of the elliptic paraboloid described by $z =2x^2 + y^2$ at the point $(1,1,3)$. The answer I get is $\dfrac{4\mathbf{i} + 2\mathbf{j} - ...
2
votes
1answer
40 views

Is this an ordinary differential equation?

If a differential equation contains only ordinary derivatives of one or more functions with respect to a single independent variable it is said to be an ordinary differential equation (ODE). If ...
1
vote
1answer
61 views

The expression $1 + x^2 +(-T_px+y)^2 +z^2$ is bounded below by a constant multiple of $(1+x^2+y^2+z^2)$

Suppose $T_p > 0$. Is there a simply way to show that $1 + x^2 +(-T_px+y)^2 +z^2 \geq C (1+x^2+y^2+z^2)$, for all $(x,y,z) \in \mathbb R^3$, where $C>0$.
2
votes
1answer
26 views

Domain and double integral

Let $$D = \{(x,y)\in R^2 : 0<x<y<2x,x^2+y^2>4,xy<4\}$$ and $f : D \rightarrow R$ the continus and bounded function defined by $f(x,y)=xy$ I'm stucked to find some bounds for $\iint_D ...
0
votes
2answers
72 views

What is wrong with this proof of continuity of a function of two variables?

If a function is define as: 1)$$f(x,y)=\begin{cases} \frac {2xy}{x^2+y^2} &\mbox{for} (x,y)\neq (0,0) \\0 &\mbox{for} (x,y)=(0,0) \end{cases} $$ Then the following proof argument, $$\frac ...
2
votes
0answers
29 views

To find the volume dilation, integrate the determinant of the Jacobian

On the road toward proving the change of variables theorem in several variables, is there a painless way to show that $$\text{Vol}(\phi(U))=\int_{U}|\text{det}(d\phi)|,$$ where $\phi$ is $C^1$, ...
0
votes
1answer
39 views

Is my calculation right for differentiability?(with complete resolution if right)

In the following completed example I ask if it is done right. $$f(x,y)=\begin{cases} \frac {2x^2y}{x^2+y^2} \mbox{for} (x,y)\neq (0,0) \\0 \mbox{for} (x,y)=(0,0) \end{cases} $$ Now and the partial ...
1
vote
1answer
25 views

Find the angle of intersection of the plane $4x+4y−1z=0$ with the plane $−4x−2y+3z=0$.

Find the angle of intersection in radians of the plane $4x+4y−1z=0$ with the plane $−4x−2y+3z=0$. Attempt: Write $\overrightarrow{n_1} = (4,4,-1)$ and $\overrightarrow{n_2} = (-4, -2, 3)$ and ...
0
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0answers
17 views

Structure of the Jacobian of an average pairwise distance matrix

let $X_1, \ldots, X_M \in M^{N \times 3}(\mathbb{R})$. Each $X_i$ represents the coordinates of $N$ points in $\mathbb R^3$. Further, let $d(X_i) \in M^{N \times N}(\mathbb R^+)$ be a pairwise ...
0
votes
3answers
37 views

Derivative of Unknown Function

Let's say I have a function $c(t), t \in \mathbb{R}$ and I don't know anything about it other than it is a function of $t$. If I derive said function with ... $x$ for example, what is the result? ...
0
votes
1answer
28 views

second derivatives of a multivariate function

A function f(x, y) is called Morse if all its critical points are nondegenerate. A function f(x, y) is called harmonic if the equation $f_{xx}$ +$f_{yy}$= 0 holds for all x, y. Prove that a harmonic ...
1
vote
1answer
34 views

Line integral exercise

Let $f:[-1,1] \to \mathbb R$ be a $C^1$ function such that $f(-1)=f(1)=0$ and $f>0$ in $(-1,1)$. Knowing that the graph of $f$ is containted in the semicircle $x^2+y^2 \leq 1$, $y \geq 0$, ...
1
vote
1answer
46 views

Stokes' Theorem problem (right triangle)

I am asked to demonstrate the truth of Stokes' Theorem ($\int_T curl(\vec{v}) \cdot \vec{da} = \int_{\partial T} \vec{v} \cdot \vec{dl}$) in the following problem/case: Let $\vec{v} = x y \hat{x} + ...
0
votes
1answer
16 views

Invariance of Laplace's approximation

Suppose that $D\subset\mathbb{R}^m$ and $g(\cdot)$ is a smooth function mapping $D$ into $\mathbb{R}$ with a unique minimum at $\hat{x}$ lying in the interior of $D$. Then, the Laplace's approximation ...
0
votes
1answer
13 views

Equality of two functions, given that the product of their difference and its gradient vanishes

Let $W \subset \mathbb{R}^n$ a open convex set, and $u,v\in C^1(\mathbb{R}^n,\mathbb{R})$ such that $$(w-v)||\nabla v-\nabla w||_2=0 \ in \ W$$ Can we say that $w=v$? The problem is that there are ...
1
vote
1answer
41 views

Is f(x,y)= $(x^2+y^2)^\frac{1}{2}$ not diferrentiable?(and my try)

If you define f(x,y)= $(x^2+y^2)^\frac{1}{2}$ Then $$f^\prime_x= \frac{x}{f(x,y)}$$and $f^\prime_y= \frac {y}{f(x,y)}$ Now is used the definition of the partials on (0,0) $f^\prime_x= \lim_{h \to 0} ...
1
vote
1answer
36 views

Why such a complicated counterexample to differentiable function, which has discontinuous partial derivatives

Here a counterexample is given, that a differentiable function has not necessarily continuous partial derivatives, but I asked myself why such a complicated example is given? Would simply $$ f(x) = ...
1
vote
3answers
44 views

triple integral and limits

$$\iiint\limits_H (x^2+y^2) \, dx \, dy \, dz \\ H=\{(x,y,z) \in R^3: 1 \le x^2+y^2+z^2 \le9, z \le 0 \}$$ I'm using Spherical coordinate system: $$x=r\cos \theta \cos\phi $$ $$y=r\cos \theta ...
1
vote
2answers
35 views

Distance and absolute value differences?

My textbook: '.. the length of a vector is in many ways analogous to the absolute value of a real number.' My question: How are the length of a vector and the absolute value of a real number ...
0
votes
1answer
10 views

where stuffs like $D^pf(x):V\to Mult (V^p,W)$ is given

Can one please suggest some book where things like derivative of a function maps to multilinear form is given ? I mean like where stuffs like $D^pf(x):V\to Mult (V^p,W)$ is given (where $f:E\subset V ...
2
votes
3answers
35 views

Finding multivariable limit $\lim_{(x,y,z)\to(0,0,0)} \frac{xyz}{(x^2+y^2+z^2)^{a/2}}$

Let $a>0$. Find $$\lim_{(x,y,z)\to(0,0,0)} \frac{xyz}{(x^2+y^2+z^2)^{a/2}}$$ After playing around with this a little bit, it looks like the limit is $0$ for $a<3$ and else it is $\infty$. But ...
1
vote
1answer
37 views

Multiple integral what am i doing wrong?

$$ \iint _d (x-y)^2*{{{e^y}^+}^z} \qquad d:x-y\ge -1,x-y\le1,x+y\ge1,x+y\le3 $$ i was trying to separate it to two multiple integral integrals but i cant integrate this integral, i was trying to ...
0
votes
0answers
16 views

Quantitative modeling of biological systems (a bunch of questions that I don't know how to get started on) Applied mathematics

Document for questions: http://www.physiology.arizona.edu/sites/default/files/Physiology472572_09HW1.pdf The link above leads to the problems I've been trying to figure out for the last six hours. I ...
2
votes
3answers
57 views

Suggested textbook for Multivariable Calculus [duplicate]

I have just finished single variable differential/integral calculus (with a little bit of infinite series). I'd like to jump right into the multivariable stuff; given that I intend to major in ...
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0answers
15 views

Triple integral over a region bounded by surfaces.

The problem is to calculate $$I=\underset{S}{\iiint}x^2\;dV,$$ where $S$ is the region bounded by $x+y^2=1$, $x+z^2=1$ and the plane $zy$. In my conclusions (that follows from this reasoning and ...
0
votes
2answers
21 views

gradient vector

It says we can define gradient as the unique vector $\nabla f$ such that $Df(x)(v)=\langle \nabla f(x),v \rangle$ I don't understand how uniqueness is coming. I can prove uniqueness if it was given ...
0
votes
1answer
70 views

Smooth curve that connects two points on a sphere

I am trying to prove that there is a smooth curve that connects two points on a sphere. I want to prove this by using the Implicit Function Theorem. (I know a lot of other ways, but I want to practice ...
0
votes
3answers
21 views

Find a plane that passes through a given point and contains a given line

The given point is $P(6,0,-2)$ and the line is $x = 4 - 2t,$ $y = 3 + 5t,$ $z = 7 + 4t.$ Which can be rewritten as $L = \langle 4, 3, 7 \rangle + t \langle -2, 5, 4\rangle.$ Initially I tried to set ...
1
vote
0answers
21 views

Parametrization of a Closed Section of a Sphere

I'm trying to verify Divergence theorem for a specific vector field through the part of a sphere in the first octant. I've done the volume integral, I'm just having trouble parametrizing the surface ...
1
vote
1answer
17 views

Finding the limit of a function of 2 variables

Hello I am having a hard time understanding this assignment, I need to find the limit for the equation below as it approaches $(0,0)$, but I don't know how to do that, any help is greatly appreciated. ...
0
votes
1answer
16 views

Line integral of vector field

Let $F$ be the vector field over $\mathbb R^3$ given by $$F(x,y,z)=(4x(1+\frac{z^2}{8}),\frac{\pi}{3}\cos(\frac{\pi}{3}y),\frac{x^2z}{2}-e^{\frac{z}{3}})$$ and let $C$ be the curve parametrized by ...
0
votes
0answers
46 views

Applying Green's Theorem to a Closed Complex Contour Integral

How would one apply Green's Theorem to the following complex contour integral: $\oint_\gamma $ $\frac{u^{s-1}}{e^{-u}-1)}du$. Where $\gamma$ is the Hankel Contour (counterclockwise) and R is the ...
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votes
0answers
34 views

Testing Divergence Theorem using Spherical Coordinates

I'm trying to verify the divergence theorem using spherical coordinates for the vector field $\vec{F}=r^2cos^2\theta(cos\theta\hat r-sin\theta\hat\theta)$ through the top half of the unit sphere. ...
0
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0answers
17 views

Integrating a three variable function

Let $\Omega$ be the region in the $xy$ plane bounded by a quarter circle of radius $a$, a straight line of slope $-1$ from $(0,b)$ to $(b,0)$, and the coordinate axes. Now consider the ...
3
votes
2answers
58 views

Stokes' Theorem - Stuck with a non-elementary integral

The following is an old exam problem (Calc III). It looks simple and technical, but I end up with a difficult integral and I guess I have a mistake somewhere. We are given the vector field ...
1
vote
1answer
21 views

Why 2 equations of the form F(x,y,z) = 0 for one 3D curve

It says in my analysis 2 book that a curve is given by $F_1(x,y,z) = 0$ and $F_2(x,y,z) = 0$. Why do we need two equations of $x,y,z$ To define a curve in 3D, shouldn't one be enough?
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votes
1answer
95 views

Is it possible to make a 3dimensional parametric plot of this curve that looks lika a UFO?

The solution set of $$ y (x^2y+y^3-x^2-4y) = 1 $$ Looks like ** How would the parametric plot in 3d look like (in an appropriate intervall)?
2
votes
2answers
38 views

Question on Partial Derviatives

For function $f(x,y) = x^2 y$ The partial derivatives for $x$ is $2.x.y $. I'm new to such math equation and i'm learning them now. May i know why is it so? Thanks!
1
vote
1answer
20 views

tangent line to the graph

this is the problem, I reached pi/6 & -pi/6 each time yet the website is saying my answer is incorrect. steps i took. 1) derivative of 4t-3tant/4t+3tant 2)yeilds 4+3sec^2(x) on bottom 3) set ...
1
vote
2answers
28 views

Do these two lines intersect?

L1 = <3,4,1> + t <2,-1,3> L2 = <1,3,4> + s <4,-2,5> I'm trying to see if these lines are parallel, skew, or intersect. I've already discovered that they are not parallel. I was thinking ...
2
votes
1answer
20 views

Existence of an exponential double integral (for the probabilists: Are the $L^p$-norms of Brownian local time integrable in the space variable?)

I have encountered the following integral and, with a lot of handwaving and some identities for Gaussian integrals (see for example ...
0
votes
3answers
29 views

Find the line of intersection of the planes

The planes are x+2y+3z=1 and x-y+z=1. My guess would be to set them equal to each other, since they are both equal to 1, we could write that as x+2y+3z=x-y+z. This simplifies to 3y+2z=0, it doesn't ...
0
votes
1answer
58 views

Second order differential equation, orthogonality

A temperature field T(x, t) is determined by the following governing equation: $$\frac 1\alpha\frac {dT}{dt} = \frac {d^2T}{dx^2}$$ (Eq 1) T(x,t) can be expressed as a form of expansion of T(x,t) = ...
2
votes
0answers
18 views

Vanishing eigenvalues of Jacobian

Let $f: \mathbb{R^2}\to \mathbb{R^2}$ be a Schwartz function. If the eigenvalues of $Df$ all vanish, must $f$ be constant? It is clear that the condition on $Df$ forces $\nabla \cdot f =\text{Tr } ...
2
votes
2answers
38 views

Second derivative expression

I have $f:\mathbb R^n\to \mathbb R$ and $\gamma:\mathbb R \to \mathbb R^n$, which are both $\mathrm C^2$. Considering $g=f\circ \gamma$, how could I express $g''$, second derivative of $g$ in terms of ...
2
votes
3answers
74 views

Books on differential geometry in the cases $n=2$ and $n=3$

I'm interested in learning the differential geometry of standard, "physical" space, that is $\mathbb R^2$ and $\mathbb R^3$. The sort of problems that were studied in the 18th and 19th century... ...
3
votes
1answer
37 views

Evaluate the integral $\iiint\limits_E x^2 \,\, \mathrm{d}V$

Where E is the region bounded by the xz-plane and the hemispheres $y=\sqrt{9-x^2-z^2}$ and $y=\sqrt{16-x^2-z^2}$. This is an exercise from my professor guide. What I tried so far: These exercise ...
3
votes
1answer
115 views

Find vector field given curl

I have an equation $\nabla \times \vec{B} = \mu_{0}\vec{J}$, where $\vec{J} = \left\langle f(x,y), g(x,y), 0 \right\rangle$ and need to solve for $\vec{B}$. I've looked elsewhere on here for how to ...
1
vote
1answer
59 views

Evaluate line integral without parameterizarion

It's been brought to my attention that line/surface integrals and integrals of differential forms in general can be evaluated without introducing a parameterization, however I haven't been able to ...