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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1answer
18 views

Finding the local extrema of a function

One of our final exam exercise sheets features this particular exercise : Find the extrema of : $2xy^3+y-x^2=0$, where $y=y(x)$ . As I thought it, this exercise involves the implicit function ...
1
vote
1answer
22 views

Derivative of bilinear forms

I want to solve the following problems: Let $f:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a bilinear form. Prove that it's differential is $$ Df_{(x,y)}(a,b) = f(x,b) + f(a,y).$$ ...
0
votes
1answer
10 views

Plane equation x units from point

I'm trying to find the equation of a plane normal to a certain vector $<x_1, y_1, z_1>$, and x units from a given point, $(a,b,c)$. Normally this question would be trivial, and I would simply ...
-2
votes
1answer
25 views

Find diff. function $f:\mathbb{R}^2 \to \mathbb{R}$ such that $f(x,y) \neq 0, f(0,0) = 0$ and $|R(0,x)|/|x|^n \to 0 \forall n$. [on hold]

Give an example of a differentiable function $f:\mathbb{R}^2 \to \mathbb{R}$ such that $f(x,y) \neq 0, f(0,0) = 0$ and $|R(0,x)|/|x|^n \to 0 \ \forall n$. Here $R(x,y)$ denotes the error term while ...
0
votes
1answer
13 views

stokes theorm on intersection curve

Using stokes theorm, evaluate line integral $\int_L f.dr $ where L is intersection of $ x^2+y^2+z^2$=1 and x+y=0 traversed in counter clockwise direction when viewed from (1,1,0). f=yi+zj+xk. I ...
0
votes
0answers
65 views

Show $f:\mathbb{R}^n \to \mathbb{R}^m$, $n>m$ can't be 1-1

Problem 2-37 on p. 39 of Spivak's Calculus on Manifolds asks Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a continuously differentiable function. Show that $f$ is not 1-1. (Hint: If, for example, ...
0
votes
2answers
32 views

does anyone know how to graph $x^2+2y^2+3z^2=12$?

I just can't think of how I should draw this graph in 3 dimensions. Can anyone draw a graph for this?
2
votes
2answers
20 views

Limit of non-linear multi-variable function

I'm trying to prove the limit of the following function is $0$: $\lim_{(x,y) \to (1,-1)} {x^3} - {2xy^2} + 1$ I know that I'm trying to find a $\delta$ s.t $ 0 < \sqrt{(x - 1)^2 + (y + 1)^2} < ...
4
votes
0answers
24 views

Is there a relationship between curl and area?

The cross product of two vectors is a new vector which lies on a new direction perpendicular to the plane of the multiplicand vectors. Its magnitude is the area of the parallelogram formed between ...
0
votes
0answers
16 views

Steepest descent from saddle point

I have the function $w(z)=\frac{1}{3}z^3+z$ where $z=x+iy$, i.e. a complex number. I am asked to find the saddle points of this function and then show the paths of steepest descent are ...
0
votes
1answer
36 views

Are these sets bounded or not?

Definition: A set $M \in R^n$ is bounded if there is a number C such that $|x| \leq C, \forall x \in M$. Problem: Determine if the following sets are bounded or not. 1) $\{ (x, y, z) : x^3 + y^3 + ...
1
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0answers
19 views

Velocity and acceleration problem

Show that if the dot product of the velocity and acceleration of a moving particle is positive (or negative), then the speed of the particle is increasing (or decreasing). If at all times t the ...
0
votes
1answer
25 views

Tangent plane and normals in $\mathbb{R}^2$

Let $f : \mathbb{R}^2 \to \mathbb{R}$ be a function given by $$z = f(x, y) = x^4 + y^4.$$ Find the point on the surface $z = f(x, y)$, where the normal to the surface is perpendicular to the chord ...
1
vote
1answer
19 views

Inconclusive second derivative test at (0,0) for $x^{4} + y^{4} - 2x^{2} - 2y^{2} +4xy $

Second derivative test is inconclusive here , given f( x, y) is $x^{4} + y^{4} - 2x^{2} - 2y^{2} +4xy $ At (0,0) how do i check nature ? Also i would like to know general tactics when things like ...
1
vote
0answers
52 views

How to reach Moore-Penrose pseudoinverse solution to minimize error function

Edit I'm trying to figure the derivation of the Moore-Penrose pseudoinverse for linear regression. The starting expression is the standard error function. I'm not quite sure how to expand on this ...
1
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0answers
38 views

Prove that exists $\delta>0$ such that, if $(x,y)\in S$ satisfies $\lVert(x,y) \rVert < \delta$, then $f(x,y) \leq f(0,0)$.

This exercise appeared on my Calculus II exam, and I didn't know even how to start doing it. Any hint is appreciated. Let $\ f, \ g : \mathbb{R^2}\to \mathbb{R}$ two $C^2$functions over the plane. ...
-1
votes
1answer
26 views

Is every point of rational number boundary point?

While studying first chapter of multivariable calculus, I am wondering if every point of the rational number is boundary point. It is obvious that $\Bbb{R}^n$ is the union of interior, exterior, ...
1
vote
1answer
21 views

Expression defined by exponential random variables, probability of being nonnegative

Consider $n \geq 2$. Let $E_1,...,E_n,F_1,...,F_n$ be independent exponentially distributed random variables with rate $1$. Define $T_E = \displaystyle \sum_{i=1}^{n}{E_i}$, and $T_F = \displaystyle ...
3
votes
1answer
46 views

Volume of a sphere with two cylindrical holes.

Consider a sphere of radius $a$ with 2 cylindrical holes of radius $b<a$ drilled such that both pass through the center of the sphere and are orthogonal to one another. What is the volume of the ...
4
votes
2answers
42 views

Find that the limit is $0$

I have to prove that the following limit is $0$: \begin{equation} \lim_{(x,y)\to (0,0)}\frac{\lvert x\rvert^2y^2}{x^2+y^4}=0. \end{equation} This is a part of an exercise where I have to study the ...
0
votes
0answers
44 views

Integration: Step in paper unclear

I've seen in a paper the following step: $$2\operatorname{Re}\int_{\mathbb{R}^n} r \partial_r \bar u \Delta u \, dx=(n-2)\int_{\mathbb R^n} |\nabla u|^2$$ This is not clear to me as I calculated: ...
2
votes
1answer
28 views

If a continuous function is nonzero at a point $a$, there is a ball around $a$ in which it has the same sign as $f(a)$

Let $f$ be a scalar field continuous at an interior point a of a set $S\in \mathbb{R}$. If $f(a)\ne 0$, prove that there is an $n$-ball $B(a)$ in which $f$ has the same sign as $f(a)$. The ...
0
votes
1answer
26 views

An unusual “multivariate Gaussian integral” that comes up when trying to translate results about a standard Gaussian to the general case

I am trying to solve this question and it leads me to a strange looking integral that I do not know how to solve. Let $\Sigma$ be positive semidefinite, and $1>\lambda>0$. I am not certain I am ...
1
vote
1answer
19 views

Trick to finding length of parametric curve

I was giving the parameters of the curve: $x = 2cos(2t)$ $y = 2sin(2t)$ and $z = 1$, where $ 0 \leq t \leq 10 \pi$ This curve describes a cylinder in the $z$ direction, and seems very straight ...
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0answers
23 views

Understanding the Jacobian past calculus

What's taught in calculus: In the calculus of multiple variables I learned that the Jacobian $$\textbf ...
0
votes
0answers
10 views

Continuous scalar field at an interior point of S and same sign proof.

Let $f$ be a scalar field continuous at an interior point $a$ of a set $S \in R$. If $f(a)$ is not $0$, prove that there is an $n$-ball $B(a)$ in which $f$ has the same sign as $f(a)$. The above ...
1
vote
1answer
26 views

Double integral of $e^{3+y^2}$ over a triangle

Evaluate $\iint_{A}^{} e^{3+y^2}dxdy$ where $A$ is a triangle with vertices $(0,0)$, $(0,-1)$ and $(1,-1)$. I don't know how to bite that. I tried multiplying it by $e^{x^2}$ and then changing the ...
2
votes
1answer
47 views

Problem with Lagrange multipliers

I am asked to find local extrema of $f(x,y,z)=ax+by$ ($a,b$ non-zero and fixed) defined on $\{(x,y,z)\colon (x,y)\neq 0\}$ subject to $$\left (R-\sqrt{x^2+y^2}\right)^2 + z^2 - r^2 = 0.$$ (here ...
1
vote
2answers
74 views

Calculus of Variations: Understanding functional derivative

I am trying to understand the basics of the Calculus of Variations and the first thing to understand is the functional derivative. I failed to find a good introductory material, so I am trying to make ...
0
votes
0answers
14 views

Finding 1st,2nd and 3rd derivative for funtion of 2 variable

$E=g(p,v)$ $\frac{dp}{dv}=F$ $\frac{dE}{dv}$=$g_pF+g_v$ $\begin{align}\frac{d^2E}{dv^2}&=(g_pF+g_v)_pF+(g_pF+g_v)_v \\ &=g_{pp}FF+g_pF_pF+g_{vp}F+g_{pv}F+F_vg_p+g_{vv} \end{align}$ ...
0
votes
2answers
18 views

Why does the slope of a smooth simple closed curve have winding number one?

$\def\RR{\mathbb{R}}$Let $S^1$ be the circle and let $\gamma : S^1 \to \RR^2$ be a smooth injective map with $\gamma'(t)$ everywhere nonzero. What is the easiest way to show that $t \mapsto ...
0
votes
3answers
42 views

integrals calculation got wrong with the extra 2

Given $$ f(x, y) = \begin{cases} 2e^{-(x+2y)}, & x>0, y>0 \\ 0, &otherwise \end{cases}$$ For $ D: 0 <x \le 1, 0 <y \le2$, I'm trying to calculate this $$ \iint_D f(x,y) \, dxdy ...
-1
votes
0answers
14 views

Simplification of integral region (no integration skills needed)

We have the following "formula" or simplification for integrals: Let $f_i:[0,1] \rightarrow \mathbb{R}^{d\times d}$ for $i=1,\dots,n$ and $g_j:[0,1] \rightarrow \mathbb{R}^{d\times d}$ for ...
2
votes
3answers
55 views

How to draw a contour map of $ f(x,y)=x^2+y^2+xy$

I have used a program to see that it is an ellipse but I want to know the process of thinking to actually draw the contour map myself. $x^2+y^2+xy=C$ for $C=0,1,2,3,...$ I can't seem to get it into ...
1
vote
1answer
32 views

Is $f(x,y)=-xy$ neither concave nor convex?

Is $f(x,y)=-xy$ neither concave nor convex? I used the definition for first differentiable functions and determined it depends on the choice of points, hence it is neither.
2
votes
1answer
35 views

Change of variables in multi-variable calculus?

About the last equality, I know it is change of variables. Let $\xi=x+t,\eta=-x+t$, but I don't know how to get the integration domain? I have been thinking for an hour and I can't get the ...
2
votes
1answer
33 views

Extending Taylor's theorem from one to several variables

In my calculus class we are dealing with Taylor´s theorem in several variables. When we were looking at the function $f(x,y)=\sin(xy)$ my teacher said that instead of applying the theorem in several ...
0
votes
1answer
30 views

three elementary problems on limits of several variable . [on hold]

I'm learning limits of several variable new. Can anyone help me? Computing the following limits: $\lim_{(x,y)\to(0,0)}|x|^y$ $\lim_{(x,y)\to(0,0)}\sin(x/y)$ $\lim_{(x,y)\to(0,0)}x^2\cdot ...
1
vote
2answers
31 views

Does every ball of boundary point contain both interior and exterir points?

My question is If $x$ is a boundary point of $S$ ($S$ is subset of $R$), does every ball of $x$ contain both interior points and exterior points of $S$? I think this is false. Since $R$ is union of ...
1
vote
1answer
42 views

Let $f$ and $g$ differentiables such that $|f'(t)| \le g'(t)$, for all $t \in [0,1].$ Prove that $|f(1)-f(0)| \le g(1)-g(0)$

Let $f:[0,1] \rightarrow \mathbb{R}^m $ and $g:[0,1] \rightarrow \mathbb{R}$ differentiables such that $|f'(t)| \le g'(t)$, for all $t \in [0,1].$ Prove that $$|f(1)-f(0)| \le g(1)-g(0)$$ Comments ...
1
vote
1answer
34 views

To check if (0,0) is local minima for$F ( x, y) = x (x - 2y^{2}) $

Hello Thanks for your time $F ( x, y) = x (x - 2y^{2}) $ . I have applied second derivative test which does not give any result . By looking at function i see that when x is greater than $2y^{2} ...
-1
votes
0answers
16 views

second derivative fail, classify the nature of the critical point

f(x,y,z)= $\frac{(x+y)^2}{2}+z^3$ the critical point I calculated is span{(1,-1,0)} the eigenvalue of the Hessian of point (1,-1,0) are 0,0,2, which means that this point is degenerate and the ...
1
vote
1answer
26 views

Simplification of integration region. (Shuffle product?)

Let us define the usual $n$-dimensional simplex: $$\Delta_{a,b}^n = \{x_1,\dots, x_n\in [0,1]^n: a<x_1<\cdots <x_n<b\}.$$ Imagine we have an integral like: $$I:=\int_{\Delta_{a,b}^n} ...
1
vote
1answer
32 views

How to guess that $f(x,y)$ has no limit?

I need to determine if the limit as $\mathbf{x}\rightarrow \mathbf{0}$ exists for the following functions: ($f:\mathbb{R}^2 - \{(0,0^T)\} \rightarrow \mathbb{R}$) $f(x_1,x_2) = ...
4
votes
4answers
457 views

Find the volume of the set.

Let $$S=\{x=(x_1,x_2,\cdots,x_n)\in \Bbb{R}^n:0\le x_1\le x_2\le \cdots \le x_n \le 1\}$$ Find the volume of the set $S$. I tried writing it as a multiple integral but it got complicated.
0
votes
1answer
26 views

'Meaning' of a triple integral with $f(x,y,z)\neq 1$

I'm studying for my Calculus II exam, and this question came to my mind while I was practising integals with spherical coordinates. Probably this question doesn't have sense at all, but there's a ...
0
votes
2answers
29 views

Gradient of dot product of two vectors

I am taking a class in which knowledge of gradients is a prerequisite. I am familiar with gradients but don't have too much experience, so I am having trouble understanding the following example. ...
0
votes
1answer
16 views

The Point of Tangency Between a Sphere and a Tangent Plane

Find the equation of the sphere centered at (2,0,-3) that is tangent to the plane x=y. What is the point of tangency? Describe the interior of the sphere with an inequality. What I have thus far: ...
1
vote
0answers
35 views

What is the intersection between $x + y - z = -2$ and $z^2 = x^2 + y^2$

I got the answer as $4x + 4y + 2xy + 4 = 0$ by substituting $z = x + y + 2$ into the second equation, but I feel as this is wrong since I am missing $z$ in the function. How do I approach this ...