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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10 views

Why is the parameter (t-1) in this example?

Example 1 on this page: http://mathinsight.org/parametrized_curve_tangent_line_examples Why do they use $(t-1)$ in the last step ($l(t)=c(1)+(t−1)c′(t_0)$)? Why not just use $t$?
7
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1answer
278 views

Two-variable limit, quotient of polynomials

I'm trying to evaluate the following limit, $$ \lim_{(x,y)\to(0,0)} \frac{x^3-y^2}{x^2-y} $$ which I think it doesn't exist, since for the curve $\alpha :[0,1]\to \mathbb R^2$, $\alpha(t) = (t, t^2)$ ...
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1answer
52 views

Integral over Gaussian curvature (Gauss-Bonnet)

Calculate the integral $\int_{M}KdA$, where $K$ is the Gaussian curvature and $M=\{(x,y,z)\in \mathbb{R^3}| x^{2}+y^{2}-z^{2}=1, x,y > 0\, and\, 0 < z < 2\}$. I wan't to do this with the ...
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1answer
45 views

Gradient versus Tangent

I am in calculus 3 and I have a question on gradient versus tangent. I know that this has been answered several times before: Difference between a Gradient and Tangent But I still have much trouble ...
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2answers
41 views

Calculating operator (matrix) norms using eigenvalues?

A remark that went unproven in class. It was said that the operator norm of a real linear transformation (real matrix) is the square root of the abs value of the max eigenvalue of $A^T*A$ (or maybe ...
2
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1answer
59 views

Vector Integration - Intuition

I understand that an integral of a scalar valued function can be visualized as "signed area under the curve". But what about integration of a vector valued function by its parameter? Is there a ...
0
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1answer
19 views

Double integral problem: Substitution where the region is defined as inequalities.

I am having trouble with the following problem: Find the area of the region $R=\{(x,y)|x\leq0 , e^{x} \leq y \leq e^{\frac{1}{2} x} \}$, by means of the substitution $x=\ln(u^2 v) $, $ y=uv$ for ...
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3answers
306 views

Show that $g(x,y)=\frac{x^2+y^2}{x+y}$ is continuous at $(0,0)$.

Let $g:\mathbb{R^2}\rightarrow \mathbb{R} $ so that, in $M=[0,1]\times[0,1]$, $$g(x,y)=\begin{cases}\frac{x^2+y^2}{x+y} &\text{ if }x+y \neq 0,\\\\ 0&\text{ if }x+y=0\end{cases}$$ Show that ...
1
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2answers
373 views

Using summation notation to prove the Leibniz rule for the gradient of product

Use summation notation to prove that $\vec\nabla (fg) = f \vec\nabla g + g\vec\nabla f$ where $f$ and $g$ are scalar functions. So I'm assuming that I need to start by writing this in the ...
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1answer
286 views

Finding tangent plane at a point to the surface

Z is a surface satisfying $$ e^{xz} \sin (y) =yz$$ The tangent plane at point $$ (0, \pi/2, 2/\pi) $$ I know that the tangent plane will be given by $$ z - c = f(a, b) + f_x(a,b) (x-a) + ...
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1answer
26 views

How to prove $\int_{B(0,1)}\frac {1}{log\left( 1+\frac{1}{|x|}\right) }dx=C\int_{0}^{1}\frac {1}{\left[log(1+\frac{1}{r})\right]^n}\frac{1}{r}dr$

I was reading a textbook, and in the proof of one of the theorems; the author claims the following without providing the intermediate steps that lead him to make such a claim: $$\int_{B(0,1)}\frac ...
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0answers
15 views

Is the $r$-axis in spherical coordinates the same as the $z$-axis?

If Cartesian coordinates have an $x$-axis, $y$-axis, and $z$-axis, do spherical coordinates have an $r$-axis, a $\theta$-axis, and a $\phi$-axis? Since the Cartesian $z$-axis is just the set: ...
2
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1answer
33 views

$C^2$ function on closed domain has finite solutions

Let $\Omega $ be a bounded open set of $R^n$ and $f\in C^2(\overline\Omega)$. Define $N_f=\{x|x\in\Omega,J_f(x)=0\}$, where $J_f$ is the Jacobian of $f$. How can I show that For all $ p\notin ...
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2answers
44 views

How do I take the derivative of this integral?

Let $$r(x,y)=\begin{cases} y &\mbox{ if } 0\leq y\leq x \\ x &\mbox{ if } x\leq y\leq 1\end{cases}$$ Show that $v(x)=\int_0^1r(x,y)f(y) \ dy$ satisfies $-v''(x)=f(x)$, where $0\leq x \leq 1$ ...
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2answers
56 views

Calculating $\iint (x+y) \, dx \, dy$

By using the change of variable $u=x+y$ , $v=x$ evaluate $$\iint_{Ta} (x+y) \, dx \, dy$$ where $Ta$ is the region in the $xy$ plane bounded by the $x$ and $y$ axes and the line $x+y = a$. Update: I ...
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0answers
82 views

partial derivative of a facet normal wrt to one of its vertex

I am struggling to understand the derivation of an equation in a paper (A Bayesian Method for Probable Surface Reconstruction and Decimation, specifically Eqn. 16). Basically they define three ...
0
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0answers
23 views

Taylor coefficients of a function

I'm having some trouble trying to prove the following: Prove that in the Taylor Polynomial of $\:f(x,y)= \sin(xy)$, centered in $(0,0)$ just the coefficients of order $4k-2$ are nonzero, for $k ...
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1answer
25 views

Multivariable calculus, stuck finding potential functions?

I am having a hard time with a question on my calc 3 hw. Find a potential function for $F(x, y, z) = \langle y + z, x + z, x + y\rangle$ on $\Bbb R^3$. I've determined it is conservative.
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0answers
64 views

How to use cylindrical coordinates to find volume of solid bounded by sphere and cone?

Integrate the function $$f(x,y,z)=5x+5y$$ over the solid given by the figure below (the bounding shapes are planes perpendicular to the x-y plane, a cone centered about the positive z-axis with vertex ...
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2answers
15 views

Prove that this map is a conservative field.

Let $h:\mathbb{R}\to \mathbb{R}$ be a continuous function. Consider the following map $F: \mathbb{R}^2 \to \mathbb{R}^2$ defined by: $$F(x,y)=\left(y^2h(xy^2),2xyh(xy^2)\right).$$ Prove that this ...
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2answers
249 views

Delta function and integrating over level sets?

Consider the three-dimensional integral $$ \int_{\mathbb R^3} d^3x\,f(x)\delta(g(x)) $$ where $\delta$ is the dirac delta, $f,b:\mathbb R^3\to\mathbb R$ and $g(x) = 0$ on some surface $S$. Is there ...
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1answer
94 views

Continuity of a function with 2 variables x,y given by $f(x,y)= (1+xy^2)^{\frac{1}{x^2+y^2}}$

i ran into this question: check if this function is continuous: $f(x,y)= (1+xy^2)^{\frac{1}{x^2+y^2}}$ when $(x,y) \neq (0,0)$ $f(x,y)= 1$ when $(x,y) = (0,0)$ thanks in advance, yaron.
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2answers
507 views

Continuity of multidimensional function: $f_1(x,y)=\frac{x^2y}{x^2+y^2}$ and $f_2(x,y)=\frac{2xy^3}{(x^2+y^2)^2}$

I am wondering how to check whether a multidimensional function is continuous. I.e. I am thinking of functions like $f_1(x,y)=\frac{x^2y}{x^2+y^2}$ with $f_1(0,0)=0$ ...
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2answers
97 views

Finding multivariable limits for the function $\frac{3x^2y}{x^2+y^2}$

Could anyone help me with this Find $$\lim_{(x,y) \to (0,0)} \frac{3x^2y}{x^2+y^2}$$ if this limit exists I tried using the squeeze theorem, but i could not find a suitable expression for the ...
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0answers
32 views

Reference request for examples of integration of differential forms over manifolds

I am studying integration of differential forms over differentiable manifolds and I would like some reference where I can find examples of actual calculations illustrating the generalized Stokes' ...
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1answer
57 views

What are all angle preserving linear operators on $\mathbb R^n$?

I´m working on Spivak's Calculus on Manifolds and I met this exercise. My immediate answer was 'all the rotations' but I can't explain why. Am I right? Can you give a hint or something to be able to ...
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2answers
85 views

What is the potential function of the field $\left(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}\right)$

The vector field is obviously conservative on every closed domain that doesn't encompass the point $(0,0)$, so there must be a potential function. I've got $\arctan(\frac{x}{y})$ for $x$ unequal to ...
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1answer
376 views

Is $f(x,y)=\frac{xy^2}{x^2 + y^2}$ continuous at $(0,0)$

$$ f(x,y) = \begin{cases} \frac{xy^2}{x^2 + y^2} & \text{ if } (x,y) \neq (0,0) \\ 0 & \text{ if } (x,y) = (0,0)\end{cases} $$ (i) Is $f$ continuous at $(0,0)$? At $(x,y) \neq (0,0)$ this ...
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3answers
763 views

Proving that the function $\frac{x^2y}{x^2 + y^2}$ is continuous at $(0,0)$.

How would you prove or disprove that the function given by $$f(x,y) = \begin{cases} \frac{x^2y}{x^2 + y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases}$$ is continuous at $(0,0$)?
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2answers
226 views

Proving that $f(x,y) = \frac{xy^2}{x^2 + y^2}$ is a continuous function using epsilon-delta.

THE QUESTION: Use the metric $(x,y)$ = $\rho(x,y)=|x-y|$ for the reals and use the metric $\rho((x_1,y_1),(x_2,y_2)) = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$ for the plane. Define $f:R\times R \to R$ as ...
0
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1answer
22 views

Composing an iterated double integral given the equation of three lines which form a triangle.

I am being asked to compose two iterated double integrals given a specific region. I have to compose a type I and a type II iterated integral for the region (I do not need to evaluate the integral, ...
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1answer
16 views

Surface integral as integral of secant

In this pdf I found looking for a solution to another problem I had earlier I see that the author claims that $$\iint dA = \iint\sec(\gamma)dydx$$ Could someone explain where this formula comes from? ...
3
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1answer
74 views

Find absolute maximum and minimum of a function with two variables

I have the function$ f(x,y)=(x+y-2)^2$ I have the constraints $0\leq x \leq 3$ and $x \leq y \leq 3$ The partial derivatives are $f_x =2(x+y-2)$ and $f_y =2(x+y-2)$ So the stationary points in the ...
2
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3answers
60 views

Calculate $\int_{\mathbb{R}^3} e^{-\left \| x \right \|}d^3x$

I've tried to find and similar question like this but I couldn't. So, I need to calculate the following integral: $$\int_{\mathbb{R}^3} e^{-\left \| x \right \|}d^3x$$ I need a hint to proceed...
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4answers
4k views

Calculator similar to Desmos but for 3D

Is there a calculator with functionality similar to Desmos but in 3 dimensions? I am looking to learn about families of quadric surfaces so I am looking for a 3D calculator with sliders.
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1answer
33 views

Multivariable calculus, taking derivative of composite functions! Help please

So I am given the following: $$f(x,y)=x^y$$ $$u(x,y) = x + \ln y$$ $$v(x,y) = x - \ln y$$ and suppose that a new function defined as: $$g(u,v)=f(x(u,v),y(u,v))$$ and I am asked to find the partial ...
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1answer
14 views

Let $S\subset R^3$ a surface described by the function $f(x,y,z)=1$ Prove that the vector $\vec {OP}$ is normal to the surface $S$

Let $S\subset R^3$ a surface described by the function $f(x,y,z)=1$ where $f$ is a $C^1$ function.Suppose that $P$ is farthest point from the surface to the origin. Prove that the vector $\vec {OP}$ ...
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1answer
85 views

Multivariable calc “second course” that does differential forms

I've worked through a computation-heavy, "standard" but quite nonrigorous treatment of multivariable calculus in the past. What book would do well as a rigorous (but not overly) "second course"? In ...
0
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1answer
36 views

Find a particular solution to the differential equation

I'm trying to solve the following multiple choice question, but I'm not getting the right answer. To find a particular solution of the inhomogenous differential equation $$(D-1)^2 (D-2)(D^2+1)y = ...
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0answers
51 views

Find average value of function over tetrahedron

The question is: Find the average value of $f(x, y, z) = x + y + z$ over the tetrahedron with vertices $(0, 0, 0)$, $(1, 0, 0)$, $(0, 1, 0)$, and $(1, 1, 1)$. I know how to find average value but am ...
2
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3answers
69 views

Gradient of $x^T B^T B x - x^T B^T b - b^T Bx$

I want to compute the gradient $\nabla_x f(x)$ of $f(x) = x^T B^T B x - x^T B^T b - b^T Bx$ with respect to the vector $x$. So far I have tried below. But when I try to add them together, I couldn't ...
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2answers
33 views

divergence free vector fields on non-simply connected domains

We know that divergence free vector fields are themselves curls of vector fields on simply connected domains. I want to construct a counter example in the case the domain is not simply connected. So ...
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0answers
13 views

Is this a function composed of multiple functions, i.e a function of a function? and how to solve it! //multivariable analysis

So i am given the following: $f(x,y)=x^y$ $u(x,y)=x+lny$ $v(x,y)= x-lny$ and suppose that a new function defined as: $g(u,v)=f(x(u,v),y(u,v))$ and i am asked to find the partial derivative at a ...
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1answer
39 views

Why direction of the curve does not matter in linear integral in respect to arc?

So i know that the linear integral in respect to arc is the area and no matter which direction we go with the curve we will get the same result. But could anyone show me a proof that does not rely on ...
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0answers
18 views

Finding parameter from inverse relations

$ x,y,z$ are known functions of $t$ and $f$ is a known function: $$ z(t)= f ( x(t),y(t)) ; $$ In order to compute $t$ as a function of $x$(or other dependent variables) how to define some ...
2
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1answer
105 views

Why are the coefficients of the equation of a plane the normal vector of a plane?

Why are the coefficients of the equation of a plane the normal vector of a plane? I borrowed the below picture from Pauls Online Calculus 3 notes: ...
25
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5answers
11k views

What is the 'implicit function theorem'?

Please give me an intuitive explanation of 'implicit function theorem'. I read some bits and pieces of information from some textbook, but they look too confusing, especially I do not understand why ...
2
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1answer
307 views

How to compute time ordered Exponential?

So say you have a matrix dependent on a variable t: $A(t)$ How do you compute $e^{A(t)}$ ? It seems Sylvester's formula, my standard method of computing matrix exponentials can't be applied here ...
0
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2answers
155 views

Continuity of a certain vector field

Let us define $$\boldsymbol{E}(\boldsymbol{x}):=\lim_{\varepsilon\to 0}\int_{D\setminus ...
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1answer
57 views

$f \in C(\mathbb R^n)$ be such that for some positive integer $m$ , $\Delta^m_{i,h}f=0 $ ; then $f$ is a polynomial in $n$ variables ?

Let $f \in C(\mathbb R^n)$ be such that for some positive integer $m$ , $\Delta^m_{i,h}f=0 , \forall h \in \mathbb R , i=1,2,...,n$ ; then how to show that $f$ is a polynomial in $n$ variables ? Here ...