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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2answers
54 views

Prove $f$ is Lipschitz on $K$

Let $f:\mathbb{R}^d\to \mathbb{R}$ such that it's partial derivatives are continuous. Let $K\subseteq \mathbb{R}^d$, a bounded set. Prove that $f$ is Lipschitz on $K$. My work: Since $f$'s ...
1
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1answer
25 views

Manifolds: show that this map is not a coordinate patch

Let $S^1$ be the subset of $\mathbb{R}^2$ given by {$(x,y)|x^2+y^2=1$}. We all know that $S^1$ is a 1-manifold in $\mathbb{R}^2$. I'm trying to prove that the following map:$$\alpha:[0,1) \to ...
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1answer
30 views

Stoke's Theorem Application on Cylinder

This is a question regarding Stoke's theorem's application. This is in regards to a problem from MIT OCW. My question is, referring to the answer provided, what closed surfaces are used in the proof ...
0
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1answer
63 views

Particular $f \in C_c^1$

is there a way to construct a function $f \in C_c^1( B(0,\frac{3R}{4}))$ such that $f|_{B(0,\frac{R}{2})}=1, \quad f|_{B(0,\frac{3R}{4})^C}=0$ and $0\le f\le 1$ everywhere such that $|Df(x)| \le ...
0
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0answers
21 views

Find all $t$ where coplanar

What are all real $t$ such that the vectors $a=<1,2,3>, b=<2,5,8>, c=<1,1,t>$ are coplanar. What I thought: I thought to find when $a\cdot (b \times c)=0$ However, the $t$ is ...
0
votes
1answer
176 views

Angle between diagonals of two faces on a cube

What is the angle between diagonals of two faces on a cube originating at the same vertex? What I have done: Vector representations of the diagonals joining the points $(0,0,0)$ to $(1,1,1)$ and ...
1
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0answers
24 views

Inverse functions from $\mathbb{R}^k$ to $\mathbb{R}^n$

Given the definition of a manifold, how do we define the inverse of the coordinate patch, or, more generally, of this kind of bijective functions?$$f:\mathbb{R}^k\to \mathbb{R}^n$$ It suffices to ...
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1answer
66 views

Comparing Two Multilinear Polynomials based on Multivariable Taylor Expansion

Given two linear functions $f(x)$ and $g(x)$ defined on real values, let's say that I want to show that $f(x) > g(x)$ for all real $x > t > 0$. According to the order-1 Taylor expansion at ...
6
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1answer
147 views

How to calculate the area of a region with a closed plane curve boundary?

Under the conditions of Green’s Theorem, the area of a region $R$ enclosed by a curve $C$ is $$\oint_C x \, dy=-\oint_C y \, dx=\frac{1}{2}\oint_C (x \, dy - y \, dx)$$ I tried to use the result to ...
1
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3answers
444 views

Volume of the region outside of a cylinder and inside a sphere

The cylinder is $x^2 +y^2 = 1$ and the sphere is $x^2 + y^2 + z^2 = 4$. I have to find the volume of the region outside the cylinder and inside the sphere. The triple spherical integral for this ...
1
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1answer
226 views

Point on surface where tangent plane is perpendicular to line.

I'm given the surface $ x^3-2y^2+z^2=27 $ and have to find where the tangent plane is perpendicular to the line described by \begin{align*} x &= 3t-5 \\ y &= 2t+7\\z&=1-t\sqrt2\end{align*} ...
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0answers
35 views

Proving multi-variable differentiability using the limit definition

I'm doing advanced calculus and I find it challenging to solve multi-variable limits while proving differentiability, more specifically 2 variable limits. could you show me how do I solve this limit?: ...
4
votes
3answers
108 views

How do one rigorously prove that the electric potential energy of an conducting sphere with charge $Q$ is $\frac{Q^2}{8\pi\epsilon_0R}$

How do one rigorously prove that the electric potential energy of an conducting sphere with charge $Q$ is $\frac{Q^2}{8\pi\epsilon_0R}$? Is integration the only way? Homogeneous charge distribution ...
1
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2answers
102 views

On proving the total differential.

I am following an open-course on multi variable calculus provided by MIT taught by Denis Auroux. The question I am about to ask is from this lecture. In the lecture Denis Arnoux gives a sketch proof ...
0
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1answer
53 views

Verifying Green's Theorem

If we have the line integral of F=$(x^2-2xy)dx+(y^2-x^3y)dy$ over a square with vertices at $(0,0)$ , $(2,0) ,(2,2) ,(0,2)$ I get the answer $24$ when doing the double integral in Green's theorem , ...
2
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1answer
219 views

Numerically find a potential field from gradient

I know that the gradient of a potential field/scalar field is a vector field, and given the function of the gradient I know how to integrate each component to get back the original scalar field. But ...
8
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3answers
285 views

Solve this functional equation:

Functional equations such as this one appear only once every several years on exams, so I feel it's hard to have a sure-fire way to approach the problem, unlike, say, solving a series convergence ...
0
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1answer
30 views

What is $\nabla\cdot A\nabla u$ for $A\in C^1(\mathbb{R}^n\to\mathbb{R}^{n\times n})$ and $u\in C^2(\mathbb{R}^n\to\mathbb{R})$?

Let $A\in C^1(\mathbb{R}^n\to\mathbb{R}^{n\times n})$ and $u\in C^2(\mathbb{R}^n\to\mathbb{R})$. How can we compute $\nabla\cdot A\nabla u$? I assume we need to apply some kind of product rule, but I ...
2
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1answer
30 views

What is $\nabla Au$ for $A:\mathbb{R}^n\to\mathbb{R}^{n\times n}$ and $u:\mathbb{R}^n\to\mathbb{R}$?

Let $A:\mathbb{R}^n\to\mathbb{R}^{n\times n}$ and $u:\mathbb{R}^n\to\mathbb{R}$. How can we compute $\nabla Au$? I assume we need to apply some kind of product rule, but I wasn't able to figure out ...
11
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1answer
169 views

Reducing multi-variable functions to a composition of 1- or 2-variable functions

There are some special functions of 3 or more complex variables that are analytic in some domain (a region in $\mathbb C^n$) with respect to each variable. To give some examples: the incomplete beta ...
0
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1answer
55 views

Manifold with boundary given as the pre-image of a subset of $\mathbb{H}^n$

Let $f \colon \mathbb{R}^{n+k} \to \mathbb{R}^n$ be a function of class $C^r$ for $r>1$. If $M = f^{-1}(0)$ and $0$ is a regular value of $f$, then we know (using implicity functions theorem) that ...
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1answer
36 views

How to see that this map is not smooth?

Let $f(x,y) = x^2$ and define $g(x,y) = {x \cdot f(x,y) \over x^2 + y^2 } = {x^3 \over x^2 + y^2}$. In this book here it is stated (on page 86) that this function does not extend to a smooth function ...
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1answer
118 views

Multivariable optimization- Nature of critical points when det of hessian matrix = 0

I'm struggling a bit with my multivariable optimization. Assuming the determinant of the hessian matrix ≠ 0 I have no issue, though when the det = 0 I get stumped. Example- $$f(x,y)=x^4+y^4-(x+y)^2$$ ...
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1answer
22 views

Help describing a graph for vector valued functions

Doing a little summer study and my textbook doesn't have much answers so thought I'd ask here. The topic is an introduction to vector-valued functions The question asks to 'describe the graph' $\vec ...
2
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5answers
131 views

$f,g$ diffirentiable function at point $(x_0, y_0)$ how to show that $fg$ diffirentiable function at point $(x_0, y_0)$?

I guess there is pretty simple way of showing the statement below.. I tried using definition but it seem complicated. Suppose $f, g: \Bbb R^{2} \to \Bbb R$. Prove if $f, g$ are differentiable at ...
9
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1answer
330 views

Volume of the intersection of two cylinders

I have two infinite cylinders of unit radius in $\mathbb{R}^3$, whose axes are skew lines. Say that the axis of one is centered on the $x$-axis, and the axis of the other is determined by the two ...
1
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1answer
79 views

Separate real and imaginary part of $j \cos (z)$

Given the following expression $$w = j \cos \left[ \displaystyle \frac{1}{n} \arccos \left( \frac{j}{\epsilon} \right) + \frac{m \pi}{n} \right] = j \cos (z)$$ (which is related to this question; ...
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1answer
27 views

Multivariable calculus: what principle is this step based on?

The background is that I was asked to solve the following problem using Green's formula $L$ is a Jordan curve (smooth and closed) which encloses the origin point in $xOy$ plane. Caculate this ...
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2answers
53 views

Finding the derivative of an integral with variable limits: ${\mathrm{d} \over \mathrm{d}x}\int_{x}^{x^2}{1 \over -2y}e^{-5xy^{2}}\mathrm{d}y$?

How do you compute the derivative $${\mathrm{d} \over \mathrm{d}x}\int_{x}^{x^2}{1 \over -2y}e^{-5xy^{2}}\mathrm{d}y$$ where the integral has variable limits?
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2answers
64 views

Problem with understanding a Differential in Multivariable Calculus

I have just started with Partial Differentiation and the book from where I'm learning (Mathematical Methods in the Physical Sciences) had the following problem on approximations using differentials ...
5
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1answer
69 views

double integral problem $\iint e^{\frac{x}{x+y}}dxdy$

I'm trying to integrate $$\iint e^{\frac{x}{x+y}}dxdy$$ where $y \leq (1-x)$ and $0 \leq x,y \leq 1$. I tried to define new variables as $u=x$ and $v=x+y$, but I can't solve this either. I have ...
0
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1answer
44 views

Why does the matrix product of jacobian of coordinate transformation and jacobin of reverse coordinate transformation equals the identity matrix

Why does the matrix product of jacobian of coordinate transformation (J) and jacobin of reverse coordinate transformation (J') equals the identity matrix(I)?
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1answer
40 views

how x y and z become equal in this solution?

I'm trying to understand an example given in my book but not able to understand it as I am quite weak in mathematics. In the below images I don't get how x, y and z become equal to each other. Please ...
3
votes
0answers
148 views

Proving injectivity of a multivariable function

Let I denote the interval $(0,\infty)$, we define the function $f:I^2\to I^2$ by, $$f(x,y)=\left({\Gamma(4x+y)\Gamma(y)\over {\Gamma(2x+y)}^2},{\Gamma(4x+y)\Gamma(2x+y)\over {\Gamma(3x+y)}^2}\right)$$ ...
0
votes
0answers
29 views

measurement of acceleration of an object over time

If I have a battery that weighs 26kg and this battery is placed in a car and the car hits a 15 cm bump (Angle 20 degrees) (to slow down the car), is it possible to calculate the speed of the car ...
4
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0answers
80 views

Integral of a function with an exponentiated inner product

Let $f:\Bbb R^n\to \Bbb R^n$ be a continuous function such that $\int_{\Bbb R^n}|f(x)|dx\lt\infty$. Let $A$ be a real $n\times n$ invertible matrix and for $x,y\in\Bbb R^n$, let $\langle x,y\rangle$ ...
2
votes
2answers
169 views

Does this weird series converge?

$\sum_{n\in S}$$\frac{1}{n}$, where S consists of those positive integers whose decimal expansion does not contain the digit 1. This was a part(b) question. Part (a) was an evaluation of the ...
1
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2answers
45 views

Given two curves, find parametric curve

I am given two graphs x versus t and y versus t and I have to determine the parametric curve. The two graphs I am given: Parametric curve (that is the right answer): So the solutions say that: ...
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0answers
40 views

What does $\frac{d^k h}{dx^k}$ mean in the context of vectors and regularization in machine learning?

I was watching a machine learning videos from the caltech course CS 156 and they have a slide where they talk about how radial basis functions (RBFs) can be derived from the following variational ...
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1answer
46 views

Vector notation and tangent planes!

In my calculus textbook there is a part explaining tangent planes to surfaces, and how to form the normal vector for such a plane. I get the idea (taking the cross product of the tangential vectors) ...
1
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1answer
79 views

Find min/max values of $f(x,y)=x^3-y^3+3xy$ in the set $K=[0,4]\times[-4,0]$

Find the biggest and the smallest values of the function $f(x,y)=x^3-y^3+3xy$ in the set $K=[0,4]\times[-4,0]$. So using partial derivatives we find that the critical points are $(0,0)$ and $(1,-1)$. ...
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0answers
32 views

Multivariable chain rule heat equation

here's the problem: the equation $\frac{\partial T}{\partial t}=\frac{\partial^2 T}{\partial x^2}$ should be transformed using $\xi=\frac{x}{\delta(t)}$ and $T(x,t)=\phi(t)F(\xi,t)$. The result of the ...
0
votes
1answer
61 views

rigorous proof in first year multivariable calculus

Hi could anyone help producing a proof for the following? If $f(x,y)$ is continuous on a closed and bounded region $R$, then $f$ has both an absolute maximum and an absolute minimum on $\mathbb{R}$. ...
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1answer
47 views

Every convex function is locally Lipschitz ($\mathbb{R^n}$)

I know that if $f$ is convex function so $f$ is continuous. And I know too that partial derivatives exists. What can I do?
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0answers
39 views

Is this function differentiable in $(1,-1)$?

I have this function: $$f(x,y)=\begin{cases}\displaystyle\frac{x \sqrt{x^2+y^2-1}}{x^2+y^2} &\text{if $x^2+y^2-1\geq0$}\\0 &\text{if $x^2+y^2-1\leq0$} \end{cases}$$ I have to say if it ...
2
votes
1answer
50 views

How to calculate $\nabla \ln(|\mathbf{u}-\mathbf{v}|)$

I need to calculate:$$\nabla \ln(|\mathbf{u}(\alpha)-\mathbf{v}(\gamma)|)$$where $\mathbf{u}$ and $\mathbf{v}$ are vector valued functions with $\alpha$ and $\gamma$ as independent values and ...
1
vote
0answers
375 views

Find the equation for the level surface of the function through the given point.?

$$f(x,y,z) = e^{x^2+y^2-z}, (1,4,11)$$ I keep getting $e^6 = e^{x^2+y^2-z}$, Which is the same as $6 = (x^2+y^2-z)$. But the answer key keeps telling me it is $x^2+y^2-z = \ln(6)$. Makes no sense; ...
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0answers
47 views

How do I algebraically change the integration limits of double integrals when substituting variables without looking at the plot?

So, lets say I have a pretty regular 2 units radius circle: $$ x^2+y^2=4 $$ If I want to calculate the area: $$ \int_{-2}^{2}dy \int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}} dx = 4\pi $$ So let's say I want ...
1
vote
0answers
61 views

Why is the computation of the Jacobian determinant different for certain integrations?

I am used to computing the Jacobian, when, say, changing from x,y coordinates to u,v coordinates, as computing the determinant of the derivative matrix of $x_u$,$x_v$, $y_u$, $y_v$, i.e., ...
2
votes
1answer
3k views

How do I parametrize a circle that's not centered at the origin?

If the circle were centered at the origin, of radius r, then r(cos$\theta$, sin$\theta$) traverses the circle once counterclockwise, for 0 $\le$$\theta$$\le$2$\pi$. What if the circle were centered ...