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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Axis of a Paraboloid?

The elliptic paraboloid $$x = y^2 /3 + z^2 /7$$ is a bowl-shaped surface. Along which axis does the bowl open? I don't even know to get started on this question. Thank you.
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1answer
61 views

Understanding Arc Length Parameterization- Concept behind Numbers

So, my main motive for understanding this concept comes from a problem I had to solve. it reads: Find an arc length parameterization of the line segment from $(1,2)$ to $(5,-2)$ In the book I'm ...
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1answer
41 views

Verifying Stokes's Theorem

I am trying to verify Stokes's theorem if $\vec{v} = z\vec{i} + x\vec{j} + y\vec{k}$ is taken over the hemispherical surface $x^2+y^2+z^2=1, \; z>0$ I have finished the left hand size of Stokes's ...
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32 views

The value of a line-integral using Stokes' theorem

I'm trying to show that there are only three possible values of the line integral $\int_y \vec F dr$ where $y$ is a closed curve on the cylinder $x^2 + y^2 = 1$ and $\vec F = (x + y, 2yz, y^2)$. I've ...
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1answer
59 views

if partial derivatives exist at a point is then do all directional dervative exist as well?

i thought if partial derivatives exist then gradient also exist ,then all direction derivatives should also exist . is this true and if it is not then why am i wrong ? $D_u=▽.u$ where ▽ is the ...
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1answer
58 views

Problem calculating multivariable limit.

Can anyone help me calculate: $$\lim_{(x,y)\to(0,0)}x\sin\left(\frac{1}{x^2+y^2}\right)$$
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3answers
84 views

Find convergence of series …

How do we find the convergence of $$ s_n = \sin (1!) + \frac {\sin (2!)}{1!} + \frac {\sin (3!)}{2!} + \frac {\sin (4!)}{3!} +... \frac {\sin (n+1!)}{n!} $$ I was thinking of using convergence tests ...
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1answer
39 views

Bounding the average of a vector valued function

Disclaimer: I edited the question so that it fits Daniel Fischer's comments and it becomes more general. I also provide an answer myself in case anyone might be interested in the solution. Question: ...
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1answer
48 views

If $f:[a,b]\to \mathbb R^2$ has non vanishing derivative then $f(x)=y$ has finitely many solutions

I can prove this claim if the derivative is further assumed continuous, i.e. $f\in C^1$: Assume $f_i(x)=y_i$, $i=1,2$ had infinitely many solutions $t_n\in [a,b]$. By compactness, $t_n$ has a monotone ...
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3answers
78 views

Calculating volume enclosed using triple integral

Calculate the volume enclosed between $x^2 + y^2=1$, $y^2+z^2=1$, $x^2+z^2=1$ I am supposed to solve this question using multiple integral. I am not able to visualize the resultant figure of ...
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1answer
40 views

differential equation, chain rule problem

Let $u$ be a function defined on $\mathbb R^3$ such that $u(x,y,z)=\phi(\sqrt{x^2+y^2+z^2})$, where $\phi \in C^2$. (i) Show that $\Delta u +u=0$ if and only if ...
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1answer
18 views

Determine binormal vector

I keep telling myself I have to be overthinking this somewhere, but I can't see where. Question prompt: Find the binormal vector $B(t) = T(t) \times N(t)$ at $t=0 \text{ and } t=1$. $$\text{ Tangent ...
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2answers
100 views

Evans 's PDE proof

Again, I got stuck. Please help me to understand the following: What is the meaning when you change from integration over the Ball B(x,r) to the surface integration dB(x,s), with another integral ...
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1answer
39 views

divergence theorem for surface of cylinder

I need to use the divergence theorem to evaluate the surface integral $$I = \int \int F\cdot n \, dS$$ where $F= x^3 i +y^3 j +z^3 k$ and $S$ is the surface of the cylinder $x^2+y^2 =4$ between ...
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2answers
2k views

Help evaluating triple integral over tetrahedron

I have a triple integral of $\iiint xyz\,dx\,dy\,dz$ over the volume of a tetrahedron with vertices $(0,0,0)$, $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$. Normally I would just have limits 0 to 1 but that ...
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1answer
32 views

Partial derivative of the composition $z(x,y)=u(x,-y)$

I need to show this to use it in another problem: Lets say you have $u: \mathbb{R}^2 \rightarrow \mathbb{R}=u(x,y)$. Then you make a new function $z(x,y)=u(x,-y)$ I need to show that ...
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1answer
51 views

why does directional derivative move fastest along the gradient?

i have just started learning multi-variable calculus , i learned that directional derivative moves fastest along the gradient . i am not able to digest it well as for the 2-D curves that i studied a ...
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1answer
51 views

Solving PDE by Canonical form transformation

For reference, the entire equation to be solved for $u(x,y)$ is: $A= -2x^2-8xy-8y^2+42x-14y$ $B= -5x^2-20xy-20y^2+105x-35y$ $C= 3x^2+12xy+12y^2-63x+21y$ $E= 28x+56y$ $K= -14x-28y$ where ...
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1answer
111 views

Area and volume relation (multivariable calculus problem)

Let $D \subset R^3$ a region over the plane $z=0$, if $C$ is the cone of base $D$ and vertex at $(0,0,1)$, show that $Vol(C)=\dfrac{1}{3}A(D)$, where $A(D)$ is the area of the region $D$. First I ...
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2answers
38 views

Finding minimum and maximum on region

Find minimum and maximum values of $x+2y$, if $4x^2+y^2=3$. I found $x_{max}=0.05$, $y_{max}=\sqrt{2.99}$. Am I right?
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1answer
2k views

Finding the equation of a one sheeted hyperboloid

Was working on calc 3 homework assignment and couldn't find out how to solve this question. the answer doesn't matter any more,i just really want to find out how to do it since my book seems to skip ...
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1answer
33 views

A doubt about line integral as a manifold.

I'm trying do a conection between a definition that I've learned in my graduation and nowadays a definition that I've learned in my doctorate studies. Calculus Definition: Let $C$ be a curve ...
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1answer
41 views

Closed simple curve and curl of function

Let $F: \mathbb R^3\setminus \{0\} \to \mathbb R^3$. Prove that if $\operatorname{curl}(F)=0$, then the integral $\int_C F\cdot ds=0$ for every simple closed curve $C$. Is this statement true for ...
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2answers
34 views

Curl and gradient properties for $f ( r)\vec r$

I need to show that the curl of $f( r) \vec{r}$ is $0$. I think I can use this property: $$\operatorname{curl}(Av) = \operatorname{grad}(A)\times v+A \operatorname{curl}(v)$$ I have started ...
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1answer
22 views

Help with calculating the Curl of [f(r)r_vec]

I need to show that the $\partial_x [f(r)(\vec{r})]=0$ Is the way to start this by saying that $\vec{r} = (x,y,z)?$ Then $r = \sqrt{x^2+y^2+z^2}?$ I am not sure if this is the way to start because ...
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0answers
112 views

Question from Munkres Analysis on Manifolds Inverse Function Theorem Section

This is the first exercise in the section on the Inverse Function Theorem (section 8). Let $f:\Bbb R^2\to \Bbb R^2$ be defined by the equation $f(x,y)=(x^2-y^2,2xy)$. a) Show that it is one to one ...
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2answers
42 views

calculating the divergence of a vector over a function

I need to find the divergence of $$\frac{\vec{r}}{r^3}$$ I think this is the way to solve (but I would like someone to check) ...
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1answer
60 views

Is the differential equation $y'''x''+x^2 y'' +x'y'=0$ linear?

Would an equation like this be considered an ordinary linear differential equation (linear in respect to $y$)? $$\frac{d^3y}{dt^3}\dot{}\frac{d^2x}{dt^2} + \frac{d^2y}{dt^2}\dot{}x^2+\frac{dy}{dt} ...
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1answer
55 views

condition for differentiability

Lets say you have a function: $$f: \mathbb{R}^2 \rightarrow \mathbb{R^2}=((u(x,y),v(x,y)).$$ Does it follow directly from this definition: ...
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1answer
54 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 ...
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198 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} - ...
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1answer
46 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 ...
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1answer
67 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 $C>0$.
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1answer
29 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 ...
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2answers
80 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 ...
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1answer
111 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 $C^1$, ...
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1answer
43 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 ...
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1answer
101 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 ...
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48 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 ...
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3answers
54 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? ...
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1answer
43 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 ...
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1answer
47 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 $x^2+y^2 \leq 1$, $y \geq 0$, ...
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1answer
82 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} + ...
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1answer
24 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 ...
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1answer
17 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 ...
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1answer
47 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} ...
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1answer
74 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) = ...
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3answers
54 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 ...
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2answers
45 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 ...
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1answer
12 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 ...