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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1answer
26 views

Does the formula for arc length hold for other coordinate systems?

Does the formula for arc length, integration of $\sqrt{(x'(t))^2 + (y'(t))^2 + (z'(t))^2}$, hold for other coordinate systems, such as cylindrical coordinates, meaning can I compute the integral of ...
0
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1answer
17 views

Directional derivative for differentiable function

In the directional derivative formula $$\frac{\partial f}{\partial v} = \nabla f \cdot v$$ why must $v$ be a unit vector?
0
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0answers
11 views

Implicit function theorem conclusion notation?

I am working through implicit function theorem for the first time, and I have the following understanding. Given a system of $n$ equations, \begin{equation} f_i(x_1,\dots ,x_m,y_1,\dots , y_n)=0,\ \ \ ...
0
votes
1answer
49 views

How to simplify this integrand,

I am trying to compute arc length in three dimensions but am currently stuck with integrating $$\sqrt{1+ e^{-2t} + 4e^{-2t}}$$ Can I get some hints on how to simplify? I didn't combine the second ...
2
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2answers
29 views

Define function in order to be continuous at $(0,0,0)$

Define the following function at $(0,0,0)$ in order for it to be continuous at that point: $$f(x,y,z)=\frac{x\sin x+y\sin y+z\sin z}{x^2+y^2+z^2}$$ I tried using the paths: $y=m_1x$ and $z=m_2x$ and ...
0
votes
1answer
32 views

chain-rule application

Consider $f:\mathbb{R}^3\to\mathbb{R},(x,y,z)\mapsto x+y+z$ and a differentiable function $g:\mathbb{R}^2\to \mathbb{R}$. What is correctly if I want to apply the chain rule, ...
0
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2answers
29 views

Applying the chain rule correctly to $f(x,g(x))$

Consider a function $f:\mathbb{R}^2\to\mathbb{R},\; (x,y)\mapsto f(x,y)$, $g$ and $f$ continuously differentiable, $g:\mathbb{R}\to\mathbb{R}$. How to apply the chain rule on $f(x,g(x))$ correctly? ...
2
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1answer
44 views

Integration on 4th dimension unit ball

I'm trying to calculate \begin{equation*}\int _B e^{x^2+y^2-z^2-w^2}\end{equation*}where \begin{equation*}B=\{\vec x\in\mathbb R^4:||x||\le 1\}\end{equation*}is the unit ball in 4 dimensions. I tried ...
1
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1answer
26 views

Finding directional derivative in direction of tangent of curve

just something small I couldnt get. $C$ is my curve that described by intersection of two planes: $$2x^2-y^2\:=1 ,\:2y-z=0 $$ The point $(1,1,2)$ is on the curve. $n$ is the vector whos direction is ...
3
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0answers
65 views

Show that such an $f$ cannot exist

Suppose $f:\mathbb R^n\to\mathbb R$ is a scalar field, such that for a given vector $a\in\mathbb R^n$ and any $y\in\mathbb R^n-\{0\}$ we have, $f'(a;y)>0$. Show that such a function $f$ cannot ...
1
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0answers
23 views

Show that $f$ is constant on the convex set $S$

Call a set $S$ convex if whenever $x,y\in S$, then $tx+(1-t)y\in S$ for any $t\in[0,1]$. Suppose that $S$ is an open convex set in $\mathbb R^n$ and suppose that $f:\mathbb R^n\to\mathbb R$. If ...
3
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1answer
35 views

What can we conclude about the function $f$?

Let $f$ be a scalar field, $f:\mathbb R^n\to\mathbb R$. Suppose there is an $n$-ball $B(a;r)$ centered at $a$ with radius $r$ and a fixed vector $y\in\mathbb R^n$ such that $f'(x;y)=0$ for every ...
2
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0answers
38 views

What's the differences between multi variable and vector calculus

This is a conceptual question. If we use vector calculus and multi variable calculus as synonym, will it be completely wrong? If so what topics does multi variable calculus have but vector ...
0
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0answers
13 views

Type of critical value

Let $f(x,y,z) = (x-y)^2 + e^{z^2}$. Is it correct that the origin is a critical value of $f$ that is a saddle point? I get for the Hessian matrix $\begin {pmatrix} 2 & & \\ & 2 & \\ ...
0
votes
1answer
34 views

Using partial derivates to solve for constant?

Suppose that $f$ is a function of $x$ and $y$ such that (partial derivative in respect to $x$) $f_x = x+2y$ and (partial derivative in respect to $y$) $f_y = a + 3y$ where a is a constant. What does ...
1
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0answers
17 views

Comparing double integral and single integral (finding which one is larger)

Hi I have two equations $$F(a):=\int^{\bar{x}}_{a}\int^{x}_{\underline{x}}2(x-c)\, dy \, dx$$ $$G(b):=\int^{\bar{x}}_{b}(y-c)\,dy$$ where $a,b,c \in [\underline{x}, \bar{x}]$ and $a>b>c$ I ...
2
votes
5answers
73 views

How do I prove that $\lim_{(x,y)→(0,0)}\frac{1-\cos(x^2+y^2)}{\sqrt{x^2+y^2}} = 0$

$$\lim_{(x,y)→(0,0)}\frac{1-\cos(x^2+y^2)}{\sqrt{x^2+y^2}} = 0$$ I believe this is correct since I couldn't find a directional limit that won't validate this. From what I know, I have to prove that ...
2
votes
1answer
40 views

literature on advanced calculus [on hold]

I need your opinions on this particular textbook: Advanced Calculus by Robert C. Buck. In my first year in college I finished two semesters of single-variable calculus and now I'm looking for a proper ...
8
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0answers
72 views

What is the Exterior Derivative Trying to Do?

$\newcommand{\R}{\mathbf R}$ Consider a smooth function $f:\R^n\to\R$ and let $Df:\R^n\to \R^{n*}$ be the map which takes a point $\mathbf a\in R^n$ to the linear map $Df_{\mathbf a}:\R^n\to \R$. ...
0
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0answers
14 views

Finding the angle between two vectors which involves gradients

Let $f(x,y), g(x,y)$ be two functions with the following property: For every constant $x$, the function in the variable $y$ is decreasing. In addition: $$ f(x,x^2 ) =3 , \, g(2y-3,y)=4 $$ Prove ...
1
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1answer
44 views

How find $I= \int_{x=0}^{ \frac{1}{2} } \int_{y=x}^{1-x} ( \frac{x-y}{x+y})^{2}\, dy\,dx$

In $$I= \int_{x=0}^{ \frac{1}{2} } \int_{y=x}^{1-x} \left( \frac{x-y}{x+y}\right)^{2} \,dy\,dx$$ follow the change of variables on $x= \frac{1}{2} (r-s),y= \frac{1}{2} (r+s)$ and find$I$ My try ...
0
votes
2answers
24 views

Equation for Tangent Plane and Linear Approximation

I need help finding an equation for a tangent plane to the following graph at the point $(1,2,5)$: $$z=f(x,y)=x^2+2xy$$ For this question, I got $z=6x+2y-5$ as the tangent plane. Can someone verify ...
-1
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0answers
23 views

An equation to describe a bowl shape in 3D [on hold]

I was wondering how would the 3D graph of bowl shape with positive x , y , z axis be plotted and what will be its equation. I was wondering around this and I am really excited to get this equation ...
1
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1answer
22 views

conservative vector fields - need a counterexample in $\mathbb{R}^2$?

I am given with the following statement: If $\vec{F}(x,y)$ is conservative in a region $A$ and in a region $B$ , then it is also conservative in $A\cup B$ . I know the statement is incorrect, but ...
1
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0answers
50 views

Is the Schwarz inequality a special case of the Cauchy-Schwarz inequality?

Given two vectors $\mathbf{x},\mathbf{y}$ in $\mathbb{R}^n$, we all know that:$$\left | \mathbf{x}\cdot\mathbf{y} \right | \le \left \| \mathbf{x} \right \| \cdot\left \| \mathbf{y} \right \|$$ ...
1
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0answers
15 views

Trigonometric parametrization of a genus g surface?

It is possible to find functions $\phi, \psi \in \mathbb{R}[sin(x), sin(y), cos(x), cos(y)]$, so that $S^2 = \phi( [0,1]^2)$ and $\psi( [0,1]^2)$ is a torus. Is it possible find, for any genus g, ...
1
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4answers
43 views

How to use Cross Product Properites to do proof

How do I proceed with a proof for this question? Prove that: \begin{equation} (a \times b) \cdot (c \times d) = \begin{vmatrix} a \cdot c & b \cdot c \\ a \cdot d & b \cdot ...
0
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3answers
24 views

Problem about planes

Say we have $2x+3y+3z=0$ which is a plane. Does that plane have infinite dimensions (it is a 2D "object" — forgive me as I am not a mathematician — but each side has infinite length) or is it just the ...
2
votes
1answer
28 views

Example where partial derivatives commute but are not continuous.

I am looking for an example of a function $f:\mathbb R^2\to\mathbb R$ such that there is a point $x\in\mathbb R^2$ with the following properties: 1) All partial derivatives of second order exist in a ...
2
votes
2answers
22 views

Necessary condition for local maximum

Let $\Omega\subset \mathbb{R}^n$ open, bounded and let $f:\Omega\to\mathbb{R}$ be a $C^2$-function. I want to prove: Necessary for a interior maximum $x_0\in\Omega$ is that $D^2f(x_0)$ is negative ...
0
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1answer
27 views

Statement about the gradient

Let $f \in \mathcal C^1(\mathbb R^n).$ If there exists $u \in S^{n-1}$ such that $$\nabla f(x) \cdot u \geq 0 \quad\forall x\in \mathbb R^n,$$ then $f(u) \geq f(0)$. How to prove this statement?
1
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1answer
22 views

Applying chain rule on a quadratic form

I'm trying to apply the chain rule on a quadratic form: $\frac{dx^TAx}{dx}=\frac{dx^T(Ax)}{dx}=\frac{dx^T(Ax)}{dAx}\frac{dAx}{dx}=\frac{dx^T(Ax)}{dAx} A$ But I'm stuck here. I think ...
0
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0answers
19 views

Evaluating surface area

These are two exercises from Apostol calculus that I am struggling to set up the integrals correctly. The biggest problem for me is finding the correct region $T$ under the surface $S$. Compute the ...
1
vote
2answers
24 views

What is the area of the part of the surface $z=yx$ bounded by $x^2+y^2=1$?

A parametrization of the part of the surface $z=yx$ bounded by $x^2+y^2=1$ is \begin{align} x &= u \cos v \\ y &= u \sin v \\ z &= \frac12 u^2 \sin 2v, \end{align} or $$r(u,v)=u \cos v \, ...
0
votes
3answers
110 views

Books various maths subjects [on hold]

I am a Civil Engineering student and i am planning on following physics in my career.I want to be ready for the advanced undergraduate courses that i will attend to,so i need to learn Differential ...
1
vote
1answer
13 views

Line integral of vector field/Why doesn't my solution work?

The question in its entirety: Determine for which constants A & B the vector field $$\mathbb{F} = (Axln(z))\mathbb{i} + (By^2z)\mathbb{j} + ((\frac{x^2}{z})+y^3)\mathbb{j}$$ is conservative. If ...
-1
votes
0answers
19 views

How to solve this double integral problem? [on hold]

$$D: y \leq 1, x^2 \leq y$$ $$\iint_D (y+yxf(x^2+y^2))\,dx\,dy$$
0
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1answer
40 views

Show that $\|f\|^{2}$ attains a minimum value on the interior of $B$

I am looking for any help, hints, or suggestions in how to go about this problem from a previous qualifying exam. We are given a smooth mapping $f: U \rightarrow \mathbb{R}^{n}$ whose differential ...
1
vote
1answer
16 views

Need help finding tangent plane to a surface

I'm doing a Calc III homework problem, and I cannot seem to figure out what the correct solution is. $$ \text{Find the equation of the tangent plane to the surface }z = 9 y^{2} - 9 x^{2}\text{ at the ...
1
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2answers
105 views

Limits in multivariable functions

I tried to evaluate this limit but I can't see any limited function here (the limit exists). $\lim\limits_{(x,y)\to(0,0)}\frac{2x^2y}{x^4 + y^2}$ Thank you.
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0answers
23 views

Gradient and invariance under change of basis

I intuitively would be inclined to believe that the gradients $\nabla F_i$ of the components $F_1,\ldots,F_3$ of a vector field $\mathbf{F}:A\subset\mathbb{R}^3\to\mathbb{R}^3$, $\mathbf{F}\in ...
2
votes
1answer
19 views

Choosing a vector normal to a jordan curve that points “inside”

Let $\gamma=\partial K_1(0,0)$ be the circle with radius $r=1$ and origin $(0,0)$ in $\mathbb R^2$. Then for any $t_0$ we have $\gamma'(t_0)\neq \begin{pmatrix} 0 \\ 0\end{pmatrix}$. Let ...
0
votes
0answers
47 views

Is the unit square a $2$-manifold in $\mathbb{R}^2$?

I'm using the following definition of a (smooth) manifold: It's from J.Munkres "Analysis on Manifolds". This is an exercise taken from this book: Is the unit square $[0,1]\times [0,1]$ a ...
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0answers
17 views

Finding the curvature from a set of datapoints

I have a set of 1. 1-d 2. 2-d data. I want to find the curvature at each single point. Till now I was using difference technique to find out the curvature, i.e, central difference at middle and ...
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0answers
17 views

Parameterization of an ellipsoid in spherical coordinates

\begin{align*} 25x^2+16y^2+z^2=1 \\ \frac{x^2}{4^2} + \frac{y^2}{5^2} + \frac{z^2}{20^2} = \frac{1}{20^2} \end{align*} The spherical coordinates are defined as, \begin{align*} x &= \rho ...
0
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0answers
31 views

Induction on derivatives

I have troubles understanding this induction proof: Let $$g(x) = \vert x \vert^{2k+1}$$ Show by induction: $$\frac{\partial ^N g(x)}{\partial x_{i_1} \dots \partial x_{i_N})} = cx_1n \dots x_iN \vert ...
4
votes
1answer
70 views

Condition for Continuity (two variable)

I came across the following question while studying for quals. This one is from a previous qualifier. I have a few ideas (which I'll mention below), but am stuck on how to complete the problem. Any ...
1
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2answers
26 views

A Point on Curve Where Tangent is Parallel

How do I find a point on a curve, $r=\langle 2,3t,5t^3\rangle$ in which tangent line is parallel to some plane. I thought to find the derivative first, but having trouble there as well. Anyway, this ...
0
votes
1answer
36 views

Local extrema in special directions

I am looking for the extrema of a function $G(y_1,y_2,y_3,y_4)$ subject to the constraint $y_1 = y_4 + y_2y_3.$ We know that $G$ is defined if $(y_2,y_3,y_4)$ is in the cylinder $\mathbb{D} \times ...
0
votes
1answer
39 views

Application of inverse function theorem?

I am not completely sure if this a direct consequence of the inverse function theorem. Assume that we have a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ that we can write in terms of ...