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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0
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2answers
68 views

What is the limit of $f(a,b) = \frac{a^\beta}{a^2 + b^2}$ as $(a,b) \to (0,0)$?

What is the limit of $f(a,b) =\frac{a^\beta}{a^2 + b^2}$ as $(a,b) \to (0,0)$? Clearly the answer depends on the value of $\beta$. For $\beta > 0$, we can deduce via inequalities that $\lim_{(a,b)...
1
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0answers
50 views

Approximation of continuous functions by polynomials

Let $D$ be a compact set in $R^d$. Assume $f \in C(D)$ is such that $\int_D f(x) x^\alpha dx = 0$ for any multi-index $\alpha = (\alpha_1, ... \alpha_d)$. Show that $f(x) = 0$ for all $x ∈ D$. This ...
5
votes
2answers
68 views

Can anyone tell me if this is correct?

Suppose that the temperature of a metal plate is given by $T(x; y) = x^2 +2x+y^2$, for points $(x, y)$ on the elliptical plate de fined by $x^2 + 4y^2 <= 24$. Find the maximum and minimum ...
2
votes
3answers
66 views

Give an argument for $\int_{0}^{n} x^p dx \leq 1 +2^{p} + 3^{p} + \cdots+ n^{p}\leq \int_{0}^{n+1} x^p dx$

For any $n$ and $p\geq 0$ give an argument that the following is true: $$\int_{0}^{n} x^p dx \leq 1 +2^{p} + 3^{p} + \cdots+ n^{p}\leq \int_{0}^{n+1} x^p dx$$ I'm having trouble even beginning this ...
2
votes
1answer
50 views

Continuity of a rational function

I have to evaluate the continuity of a two functions at a couple of different points and I am a bit stuck. Here are the two functions: $f(x,y) = 0$ if $(x,y)=(0,0)$, $f(x,y) = \frac{x^3y^2}{3x^4+2y^2}...
1
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3answers
115 views

Find $3$ numbers whose product is $27$ and whose sum is minimal

Find $3$ numbers whose product is $27$ and whose sum is minimal I'm thinking one might have to use langrange multipliers. The answer is $(3,3,3)$, I am not sure how to get there though.
2
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1answer
47 views

Show that $-\frac{2yx^3}{(x^2+y^2)^2}$ is bounded.

Show that $-\frac{2yx^3}{(x^2+y^2)^2}$ is bounded. I'm approaching this starting with $$ \left| \frac{2yx^3}{(x^2+y^2)^2} \right| = \left| \frac{2yx^3}{x^4+2x^2y^2+y^4} \right| \leq \left| \frac{2yx^...
3
votes
2answers
66 views

How do I change the order of integration of this integral?

$$\int\limits_1^e\int\limits_{\frac{\pi}{2}}^{\log \,x} - \sin\,y\, dy\,dx$$ I don't understand how to change the order because of the $\log\,x$ as the upper bound for the inner integral How do I ...
0
votes
1answer
103 views

Use Green's theorem to calculate line integral

I've the areas $C1: x^{2}+y^2 = 1$ $C2: r = \frac{2}{\sqrt{2-cos(\theta)}} $ $R$ is the area between C1 and C2, both curves are oriented with positive direction/flow (don't really know the ...
1
vote
1answer
51 views

Finding the Differential $L \in \hom(\mathbb{R}^2, \mathbb{R})$ to show that $f$ is differentiable at the origin

Problem: Let $f: \mathbb{R}^2 \to \mathbb{R}$ and $\varphi: \mathbb{R} \to \mathbb{R}$ such that $f(x,y)=\varphi(|xy|)$ for all $(x,y) \in \mathbb{R}^2$ Show that: If $\varphi(0)=0$ and $\exists \...
0
votes
1answer
34 views

When do two equations represent the same line in $\mathbb{R}^n$

The question states that if $\mathbb U\neq0$ and $\mathbb V\neq0$ then complete the sentence : The equations $$\mathbb X=\mathbb X_o + t\mathbb U$$ $-\infty<t<\infty$ and $$\mathbb ...
0
votes
1answer
24 views

interpreting the curve of intersection

I would like to understand the idea of a 'curve of intersection' in $\mathbb{R}^{3}$. Say we are given a surface $z = x^{\frac{1}{3}}y^{\frac{1}{3}}$ and a plane $y = x$. Then the curve of ...
1
vote
1answer
86 views

Can someone provide a simple example of the “pre-image theorem” in differential geometry?

I only have a background in engineering calculus. A problem I am currently working on relates to something called a "pre-image theorem" The theorem roughly states: Let $f: N \to R^{m}$ be a $C^{\...
0
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1answer
41 views

Determining the Change in a variable as a function of change in independent variables

I have an Equation at hand: F = V/P I'd like to find out that for a given number of unit change in F, how many units of change are due to change in V and how many units of change are due to change ...
1
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2answers
57 views

Differentiate a Function with respect to a different function.

Suppose you have two functions $f(t)$ and $g(t)$. How would you generally differentiate for any real valued functions: $\frac{df}{dg}$? And also a specific case: $$f(t) = t^3 + t^2 + 1 $$ $$g(t) = (...
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0answers
21 views

Need help in computing inner product matrix for spherical coordinates

I was practicing conversion of del operators from Cartesian to Curvillinear coordinates, e.g. spherical coordinates $$\left\{\begin{matrix} x=r sin \theta cos\phi\\ y=rsin\theta sin \phi \\ z=rcos \...
0
votes
0answers
40 views

Line integrals in space interpretation

I realize that the line integral $$\int_Cf(x,y)ds$$ can represent the area under the function f along the curve $C$ but what does it represent in 3D? For instance if the curve was a helix increasing ...
3
votes
1answer
53 views

surfaces, curves and lines

Could someone please assist with the following questions: Consider $f(x,y) = x^{\frac{1}{3}}y^{\frac{1}{3}}$ and take $C$ to be the curve of intersection of $z = f(x,y)$ with the plane $y=x$. Show ...
0
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0answers
84 views

Work done along a closed curve is zero but field is not conservative. How can that be?

For example lets say we have this vector field: $$F=<xz,yz,xy>$$ Compute $$\int\int_S Fdr$$ where $s$ is part of the sphere $$x^2+y^2+z^2=4$$ that lies inside the cylinder $$x^2+y^2=1$$ The ...
0
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1answer
140 views

Lagrange multipliers

A current of $18$ amperes branches into currents $x$, $y$, and $z$ through resistors with resistances $5$, $7$, and $4$ ohms as shown. It is known that the current splits in such a way that the sum ...
0
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1answer
57 views

Chain rule with a vector

I'm trying to apply the chain rule to $\vec{r}=\vec{r}(u_1,u_2,u_3)$ to provide the expression for $d\vec{r}$ above. However I do not seem to be able to apply the 'tree diagram' mnemonic here and it's ...
3
votes
1answer
110 views

Rewrite triple integral from Cartesian to cylindrical or spherical coordinates

How to rewrite this integral in either cylindrical or spherical coordinates? (whichever is easier). $$\int_0^1 \int_0^x \int_0^{ \sqrt{x^2+y^2}} {(x^2+y^2)^{3/2}\over x^2+y^2+z^2}\,dz\,dy\,dx$$
2
votes
1answer
83 views

On the definition of critical point

Let $f:\mathbb{R}^n\to \mathbb{R}^m$ be a smooth function (or in general between two smooth manifolds). Then $p\in \mathbb{R}^n$ is a critical point if $df_p$ is not surjective. I feel confused about ...
0
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0answers
248 views

Lipschitz continuous function of a matrix

What is the definition of Lipschitz continuous function of a matrix? Suppose $f$ is a function of a matrix. The definition should be something like $$\| f(X+S) - \ f(X) \| \le \nu \|S\| $$ But I ...
2
votes
2answers
43 views

$\operatorname{curl} \mathbf{A} =0$ if and only if $A$ is conservative

Should this theorem not instead state: $\operatorname{curl}\mathbf{A}=0$ on the surface $S$ as by Stokes' theorem $\displaystyle \oint_{\gamma} \vec{A} \cdot d\vec{r}=\int_S$curl A $\cdot $ $d\vec{S}$...
2
votes
2answers
58 views

Line integral (not using Stokes theorem)

Evaluate $$\int_C Fdr$$ $$F=<-y^2,x,z^2>$$ $C$ is the curve of intersection of the plane $y+z=2$ and the cylinder $x^2+y^2=1$ I can parametrize the curve using cylindrical coordinates but I ...
2
votes
1answer
23 views

Partial derivative w.r.t. to the time of a time-dependend quadratic form

Suppose the quadratic form $$V(x(t), t) = \frac{1}{2} x^\mathsf{T}(t) P(t) x(t)$$ where $$x(t) \in \mathbb{R}^n,~P(t) \in \mathbb{R}^{n \times n},~\text{and}~P(t) = P^\mathsf{T}(t) > 0$$ i.$\,$...
1
vote
1answer
40 views

quick question on surface integrals/stokes theorem

Say if I have a cylinder with the bottom removed I have an open surface. When we apply stokes theorem (or carry out the surface integral) are we just summing over the outer surface or the surface ...
1
vote
2answers
45 views

Multivariable taylor polynomial

$$f(x, y) = e^{2x+xy+y^2}$$ Find the 2nd order taylor polynomial to the above function about (0,0) The formula is: $$P(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)+\frac 12[f_{xx}(x-a)^2+2f_{xy}(x-a)(y-...
2
votes
2answers
90 views

evaluate the double integral $\int_0^4 \int_{\sqrt{y}}^2 \sqrt{x^2+y}\, dxdy$

evaluate the double integral $\int_0^4 \int_{\sqrt{y}}^2 \sqrt{x^2+y}\, dxdy$ Hi all, could someone give me a hint on this question? I've actually tried converting to polar coordinates but i cant ...
1
vote
1answer
28 views

Surface integral (algebraic solution)

Find the area of the part of the cone $$z^2=x^2+y^2$$ that lies inside the cylinder $$x^2+y^2=2ay$$ I would like an algebraic solution. This is how I set it up: $$\int\int_Sds = \sqrt{2}\int\...
0
votes
1answer
45 views

Equality form of second order Taylor series

I am reading a book on optimization wherein a statement using Taylor's expansion is made without proof. \begin{equation} f(\mathbf{y}) = f(\mathbf{x}) + (\mathbf{y} - \mathbf{x})^T\nabla f(\mathbf{x}...
1
vote
1answer
113 views

Definition of a 2-variable function derivative

I read this definition in a book of multivariable calculus: $f(x,y)$ is differentiable at $(x_0,y_0)$ if it can be expressed as the form $$f(x_0+\Delta x, y_0+\Delta y)=f(x_0,y_0)+A\Delta x+B\...
1
vote
1answer
54 views

The number of vertices in a polytope is finite [duplicate]

I want to prove the following: Let $K$ be a convex polytope. Show that $K$ has a finite number of extreme points. I have seen the bound for the cardinality of the set of extreme points: $|E| \leq 2$...
1
vote
2answers
59 views

Multivariable limit exists?

Does the limit $$\lim_{(x,y)\rightarrow (0,0)} \frac{y^4}{x^\beta(x^2+y^4)}$$ exists for $\beta>0$? I don't think it exists but how do you prove it rigorously. Thanks
1
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2answers
259 views

How to get projection of ellipsoid onto sphere

I'm trying to get the projection of an ellipsoid onto a sphere. Depicted in the image below, I need the projection of the red ellipsoid onto the unit sphere at the origin. I have tried various ...
0
votes
1answer
75 views

definite integral of piecewise function in R2

For $t\ge 0$ let $$ f(x,t) = \begin{cases} x, & \text{if $0\le x \le \sqrt{t}$} \\ -x+2\sqrt t, & \text{if $\sqrt t \le x \le 2\sqrt t$} \\ 0, & \text{otherwise} \end{cases} $$ ...
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0answers
65 views

Definite Integral of $\sqrt{(x^2+y^2)^k+B}$

I'm trying to evaluate the integral $$ \int_{-1}^1 \sqrt{(x^2+y^2)^k+B} \, \mathrm{d}y $$ WolframAlpha doesn't return a response even for simplified versions of this, but I believe it can be ...
1
vote
1answer
43 views

The meaning of the sum of partial derivatives of a scalar function

I have a real valued function $f(x_1,\dots,x_n)$ and it is important for me to add the coordinates of his gradient. In other words, if $u$ is the vector $(1, \dots, 1)$ I want to evaluate: $$\...
0
votes
1answer
49 views

Deriving the Maclaurin series of $\int_{0}^{x} \frac{e^t+e^{-t}-2}{t^2}dt$

I've found this question to be a bit tricky: $\int_{0}^{x} \frac{e^t+e^{-t}-2}{t^2}dt$. My first thought, knowing that $e^t = \sum_{n=0}^{\infty} \frac{t^n}{n!}$, is to break the equation into three ...
0
votes
1answer
132 views

Differentiability implies continuity in $R^2$

Let F be a function from $R^2$ to $R^2$. F is differentiable at a point (a,b) in $R^2$, prove that F is continuous at this point. Can i write F(x,y)= F(a,b)+ c(x-a)+ d(x-b)+e where c,d,e are real ...
0
votes
1answer
33 views

Double integral with parametrization

Evaluate $$\int\int_D(-2x^2+2xy+1)\,dx\,dy $$ where $$D: x^2+y^2≤a^2$$ I have parametrized as follows: $$x=a\cos t$$ $$y=a\sin t$$ $$\int_0 \int_0^a(-2a^2cos^2t+2a^2\cos t\sin t+1)r\,dr\,dt $$ $$...
0
votes
1answer
32 views

Show that the sum of the oscilations is less or equal to $f(b)-f(a)$

I want to show the following: Let $ f:[a,b]\to \mathbb{R}$ be an increasing function.If $ x_1,\ldots,x_k\in[a,b]$ are different, show that $$\displaystyle\sum_{i=1}^k O(f,x_i) < f(b) - f(a).$$ ...
0
votes
1answer
180 views

Find the point on the paraboloid $z = \frac{x^2}{4}+ \frac{y^2}{25}$ that is closest to the point $(3, 0, 0)$

Find the point on the paraboloid $z = \frac{x^2}{4}+ \frac{y^2}{25}$ that is closest to the point $(3, 0, 0)$ Hi all, could someone give me a hint on how to start doing the above question?
0
votes
2answers
275 views

Finding slope at a point in a direction on a 3d surface

This is not a duplicate, I have attempted the question and am not sure why my answer is incorrect. QUESTION: The surface with equation $z = x^3 +xy^2 $ intersects the plane with equation $ 2x−2y = 1$ ...
2
votes
1answer
182 views

Showing differentiability for a multivariable piecewise function

Let $$f(x,y)=\begin{cases} xy\sin(x/y) & y\neq 0 \\ 0 & y=0\end{cases},$$ show whether $f(x,y)$ is differentiable at $(0,0)$. It seems that there are multiple ways to do this but there ...
0
votes
0answers
49 views

The formula of Eclidean distance to a hyperplane.

I have a hyperplane eqution H: "$X - Y = 0$" where $X, Y \in R^{n\times m}$. Could you tell me how to deduce the smallest Euclidean distance formula for any point ($X_0,Y_0$) to H ?
1
vote
1answer
33 views

Surface integral problem

Find $$\int_S yds$$ where $S$ is the part of the plane $$z=1+y$$ that lies inside the cone $$z=\sqrt{2(x^2+y^2)}$$ What I tried to do: I combine the two equations to get the intersected surface ...
1
vote
1answer
68 views

Limit of integral in the wrong variable

Evaluate and justify: $\lim_{y\to 0^+} \int_0^1 \frac{x\cos{y}}{\sqrt[3]{1-x+y}}dx$ Can I just apply the limit to the $y$ directly, since in regards to the variable being integrated it's just a ...
1
vote
1answer
121 views

Isothermal parameterization, Inverse of the Gauss Map

This problem is from Do Carmo's Differential Geometry of Curves and Surfaces. It is question 13 from chapter 3.5, to be specific. Suppose that S is a minimal surface without any umbilical points (...