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
104 views

Find the center of mass of soda can?

Help finding center of mass of soda can? If you represent the soda can as a right-circular cylinder radius=4 cm height =12 cm We are told to neglect the mass of the can itself. When the can is ...
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1answer
40 views

Mutlivariable Calculus: Surface Area

This was a question a students had asked me earlier today regarding surface area. Find the surface area of the hemisphere $x^2+y^2+z^2 = 4$ bounded below by $z=1$. I decided to approach ...
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2answers
57 views

Generalized forms of Curl and Divergence

The definitions I learned in my calculus courses for curl and divergence were rather, at least to me, unintuitive and seemed to work only for $\mathbb{R}^3$. I took a look on Wikipedia: "The ...
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1answer
27 views

Setting up the intergal but do not integrate

I'm having a little trouble with this problem. Let D be the solid bounded by y=x, z=1-y^2, x=0, and z=0 1) Sketch the region of integration using 2 and dimensional sketches to show the region ...
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1answer
33 views

Notation Question with Line Integrals over Vector Field

Previous Question: What is the Convention in Arc Length Parametrization? This is a follow-up question to my previous post on line integrals going in opposite directions. At first I thought I ...
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1answer
51 views

Stokes’ Theorem to find integration

Use Stokes’ Theorem to evaluate integration $c (xy \,dx+ yz\, dy + zx\, dz)$ where and $C$ is the triangle with vertices $(1,0,0),(0,1,0),(0,0,1)$, oriented counter-clockwise rotation as viewed from ...
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40 views

Surface area with double integral - how to parameterize?

Problem: Find the surface area of the part of the cylinder $x^2+z^2 = a^2$ that is inside the cylinder $x^2+y^2 = 2ay\;$ and also in the positive octant $( x \ge 0, y\ge 0, z\ge 0$). Assume $a > ...
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1answer
115 views

Find average z coordinate..Help please?

Question: Find the average z coordinate of all the points on AND within a hemisphere of radius 4 centered at the origin, and with it's base in the xy-plane. So I am assuming the function will be ...
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1answer
591 views

Definition for monotonicity of multivariate function

Is there any standard definition for monotonicity of a multivariate function? I suppose it's something like: $\forall i: x_i \leq x_i' \implies f(x_1, \ldots, x_i, \ldots, x_k) \leq f(x_1, ...
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65 views

How to find the partial derivative of a row vector?

If I have $\vec{x} = \begin{pmatrix}x_1\\ x_2\\ \vdots\\ x_n\end{pmatrix}$, how can I find the derivative of $\vec{x}^T$? This questions comes when I was trying to find the minimum of the inner ...
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1answer
42 views

Directional derivatives with given values.

At the point (1,2), the function f(x,y) has a derivative of 2 in the direction toward (2,2) and a derivative of -2 in the direction toward (1,1). Find f_x(1,2) and f_y(1,2). Find the derivative of f ...
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24 views

Finding the value of the inverse function with inverse function theorem

I am stuck by the following problem. Let $h:\Bbb R^2\rightarrow \Bbb R^2$ and $$h(x,y)= (x^2+3xy+xy^3, x^3-5y^2)$$ Let $g=h^{-1}$ near $(0,1)$. Find $Dg(0,-5)$ I showed that the inverse function ...
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1answer
51 views

Can you switch the order of the determinants when changing variables using the Jacobian?

Let say we're changing the variables and we use the Jacobian to do this. Lets say we integrate in respect to $u$ and $v$, does it matter if we set up the integral like ...
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1answer
97 views

How to find the area inside $x^2+y^2+\sin(4x)+\sin(4y)=4$ using Green's Theorem

This was a modification of a previous question I asked, except now I'm saying how to solve the area inside $$x^2+y^2+\sin (4x) +\sin (4y) = 4 $$ The equation is a simple closed shape. Here's is the ...
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1answer
40 views

What happens to other eigenvalue? - steady state heat equation

A circular plate is bounded by circles of radii $r=2$ and $r=4$, its surface is insulated and temperatures along boundaries are given by $u(2,\theta)=10\cos\theta + 6\sin\theta$ and ...
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2answers
83 views

Show that constant curvature $\kappa = 1/r$ is necessary and sufficient that the curve is a circular arc of radius $r$

We have to prove that a curve has constant curvature $\kappa = 1/r$ if and only if it is in a circular arc of radius $r$. I am confused because doesn't a helix also have a constant curvature given ...
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1answer
55 views

Evaluate Double Integral

$$ \int _0^{\sqrt{\pi}} \int_y^{\sqrt{\pi}} \sin (y^2 )\; dydx$$ Even if I change the order of integration I don't see how to get rid of this $\sin (x^2)$ which doesn't have antiderivate. It is ...
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1answer
45 views

$n$-th derivative of $\displaystyle \left(\sum_{i=1}^Nx_i\right)^n$

I have no idea how to find the $n$-th derivative of $f:\mathbb{R}^N\rightarrow\mathbb{R},x\mapsto $ $(\sum_{i=1}^N x_{i})^n$. I tried to use the multinomial theorem, as well as only the chain rule. I ...
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1answer
195 views

Proof that continuous partial derivatives implies differentiability

This is the statement of Theorem 2.8 from Spivak's Calculus on Manifolds. I'd like feedback on if this looks fine as far as a generalization to his proof goes: Theorem: If $f: \mathbb{R}^n ...
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2answers
52 views

Studying the differentiability of a function at a point $(a_{1},a_{2})$

I have a function $\ f: \mathbb{R}^2 \to \mathbb{R} $ to study: 1) It's continuity at the point $(a_1,a_2)$. 2) The partial derivative exists at $(a_1,a_2)$? 3) Are the partial derivatives ...
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1answer
59 views

Lagrange multipliers: More than one constraint

I have more or less understood the underlying theory of the Lagrange multiplier method (by using the Implicit Function Theorem). Now, I try to extend this understanding to the general case, where we ...
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3answers
81 views

limits multivariable calculus. where am i wrong with my attempt?

P : $\lim_{(x,y) \to (0,0)} f(x,y)$ where $$f(x,y) = y\sin\frac1x + \frac{xy}{x^{2} + y^{2}}$$ Text book says Limit doesnot exist . So where i am wrong with my proof below ? EDITED ATTEMPT : Or ...
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1answer
40 views

Partial derivative and change of coordinates

A colleague posted this on the door outside his office: $$\frac{\partial}{\partial(x+y)}(xy)=?$$ Trying to be helpful, I gave it a shot: $$u = x + y \\v = x - y \\ x = \frac{u+v}{2} \\ y = ...
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1answer
23 views

Double Integral Confusion..

for the double integral $\iint$$5xye^{(-x^2)}dxdy$, are you able to pull out the $5y$ and then integrate over $xe^{(-x^2)}$? For the indefinite integral I got $5y(\left(\frac12\right)e^{(-x^2)})$, ...
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1answer
22 views

Use triple to verify that a paraboloid divides a solid in two regions of the same volume, where am I wrong?

Let $S$ be the region over the $xy$ plane and inside the intersection of the cylinder $x^2+y^2=a^2$ and the plane $z=a^2$. I want to verify that the paraboloid $z=x^2+y^2$ divides $S$ into two ...
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2answers
41 views

Evaluating the following integral: $ \iiint_S \sqrt{x^2/9+y^2/4+z^2} \, dV$.

I am calculating: $$ \iiint\limits_S \sqrt{\frac{x^2}9+\frac{y^2}4+z^2} \, dV$$ Where $S$ is the region over the $xy$ plane and inside the intersection of the ellipsoid ...
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1answer
16 views

Let $f:\Bbb{R}^2\to\Bbb{R}$, with $f(x,y)=\frac{1}{6}(3xy^2-4x^2y+y^3+10)$. Find the equation at the tangent plan of f at the point $(3,2)$.

I've answered this question and I got: $f_x(a,b)=-6$ $f_y(a,b)=2$ $f(a,b)=-3$ and my answer is: $-6x+2y+9$ Just wanted to know whether I am on the right tracks here? The second part to this ...
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1answer
50 views

Why does Continuous Partial Differentiability Imply Total Differentiability?

Let $f: \mathbb{R}^d \to \mathbb{R}$ be such that the partial derivatives $\frac{\partial f}{\partial x_i}:\mathbb{R}^d \to \mathbb{R}$ exist everywhere and are continuous. Then show that $f$ is ...
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49 views

d'Alembert Solution of wave equation for open end problem

in d'Alembert Solution of wave equation for open end problem,i saw boundary condition as $u_x(0,t)=0$,but i didn't understand why this is like that,then i saw in one book as force at free end is ...
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71 views

d'Alembert Solution of fixed end

in d'Alembert Solution for fixed end semi infinite string problem with wave equation $u_{tt} = c^2u_{xx}$,we get $0= \frac{f(ct)+f(-ct)}{2} + \frac{\int_{-ct}^{ct}g(s)ds}{2c}$ where $f$ and $g$ are ...
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1answer
508 views

Show the Grassmannian is a smooth manifold (using dummy definition of smooth manifold)

We received the following problem in my Differential Geometry class: Suppose $0\leq k \leq n$ are integers. Let $G(k,n)$ be the collection of orthogonal projections $T: \mathbb{R}^n \to \mathbb{R}^n$ ...
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2answers
31 views

Why $E^2$ instead of $\Bbb R^2$

I was reading a paper earlier where whenever the author would discuss vectors in polar coordinates, he'd call the space $E^2$. He'd even give a function of vectors in that space as $f:E^2 \to \Bbb ...
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2answers
48 views

Evaluate $\int_0^1\int_y^\sqrt{y}\frac{y}{\sqrt{x^2+y^2}}\,dx\,dy$

Not sure how to simplify this and start working it out. Any help and step through would be much appreciated thanks! Ive gotten to ...
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1answer
34 views

Approximating the definition of inverse function

Let $f:D\mapsto\mathbb{R}^n$, $D\subset\mathbb{R}^m$ open, $m<n$ be a continuously differentiable function. I define $g:\mathbb{R}^n\mapsto D$ by $$ g(y)=\operatorname{argmin}_{x\in D}\|f(x)-y\|, ...
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3answers
47 views

Expanding $(1 - x + 2y)^3$ in powers of $x-1$ and $y-2$ with a Taylor series

I would like to do this. I observe that I can write $$f(x,y) = (1 - x + 2y)^3 = (2(y-2) - (x-1) + 4)^3.$$ It's easy to do this via algebra directly. However, I'm asked to do it by computing the ...
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2answers
48 views

Given a vector field $\mathbf{H}$ .Find a vector field $\mathbf{F}$ and a scalar field g, such that $\mathbf{H}$ = curl(F) + ∇(g).

Let$\;\mathbf{H}(x,y,z) = x^2y\mathbf{i}+y^2z\mathbf{j}+z^2x\mathbf{k}$. Find a vector field $\mathbf{F}$ and a scalar field g, such that $\mathbf{H}$ = curl(F) + ∇(g). I took divergence on both ...
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22 views

Showing that the surface $z = 3x^2 - 2xy + 2y^2$ lies entirely above every one of its tangent planes

The hint given is to look at the Taylor expansion at every point in space. It's not immediately clear to me how to proceed with this. If I calculate out the equation of the tangent plane at a point ...
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1answer
22 views

Partial derivatives of a $C^2$ function $f(x,y)$ where $f(x,y) = f(y,x)$ and $f(x,x) = x$

I'd like to show that the second-order term in the Taylor polynomial for $f$ centered at $(a,a)$ is equal to $$\frac{1}{2}f_{xx}(a,a)(x-y)^2.$$ By $C^2$ I mean that all of the first and second order ...
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1answer
144 views

Multiple integration questions and my attempt

P1: Find volume of portion of sphere centered at $(0,a,0)$ with radius $a$, between planes $y=0$ and $y=a$ . I changed to spherical coordinates and I calculated volume in 1st quadrant and multiplied ...
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54 views

Order of Integrating a partial derivative

I have some questions about the process involved when integrating higher order partial derivatives. I was going through a textbook on engineering mathematics on PDEs. If $\frac{\partial^2 ...
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1answer
4k views

Distance of a test point from the center of an ellipsoid

I'm trying to learn about Mahanalobis distance and I'm pretty close to getting the idea. I've learned that the distance has got a lot to do with the properties of an ellipsoid. I have understood so ...
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1answer
27 views

integral of a function bounded over an elliptic area

I've been stuck with the following integral, I know I have to use substitution, but I don't know how. Let $S=\{(x,y):\frac{x^2}{a^2}+\frac{y^2}{b^2}\leq 1\}$. Show that $$\int \int_{S} \left( ...
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61 views

Parameterizations of Lines.

Which of the following equations give alternate parameterizations of the line L parameterized by: r(t)=(1+2t)i +(2+2t)j -(1+4t)k? a. -(1+t)i-t*j+(3+2t)k b, (3-2t)i+(2-2t)j+(3-4t)k c. ...
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1answer
17 views

Show identity with function of 2 variables.

Let $f(x,y)$ - differentiable in $\mathbb{R}^2$. Let $x=aX + bY$, $y=cX+dY$. We put $g(X,Y) = f(aX+bY,cX+dY)$. Show identity : $x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} ...
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4answers
100 views

Evaluating $\lim_{(x,y)\rightarrow (0,0)} \frac{(xy)^3}{x^2+y^6}$

$$\lim_{(x,y)\rightarrow (0,0)} \frac{(xy)^3}{x^2+y^6}$$ I don't really know how to do, but I was trying to do like that: $a=x$, $b=y^2$ then I was trying to do this $$\lim_{(x,y)\rightarrow (0,0)} ...
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2answers
38 views

Triple Integrals Problem

Find the volume formed by $x^2+y^2=9$, $z=0$, and $y=3z$. I am having trouble with determining the limits of integration. $x^2+y^2=9$ describes a circle centered at the origin with a radius of ...
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20 views

How to compute multidimensional inverse Fourier transform

everybody. I am currently working on deriving solutions for Stokes flows. I encounter a multidimensional inverse Fourier transform. I already known the Fourier transform of the pressure field: ...
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0answers
36 views

Inner product between basis vectors

Given a vector field $A=A_x \hat x + A_y \hat y$. If we want to represent this vector field with respect to the polar coordinate vector fields $\hat r$ and $\hat \phi$, we'd just perform the dot ...
2
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1answer
88 views

Evaluate the surface integral from the paraboloid

Evaluate the surface integral $$\iint\limits_S xy \sqrt{x^2+y^2+1}\,\mathrm d\sigma,$$ where $S$ is the surface cut from the paraboloid $2z=x^2+y^2$ by the plane $z=1$. Is it possible for the ...
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0answers
54 views

Classify the degenerate point;

Given $$f(x,y)=y^4+y(x-1)^2-8y^2$$ Find the three critical points, use Hessian method to classify the two non degenerate points. Then ; By considering the value of $f$ along the curve ...