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
100 views

Cauchy-Riemann equations and harmonic functions

I have the following question which I am supposed to use the Cauchy-Riemann equations to prove: Let u and v have continuous second derivatives satisfying: $$\frac{\delta u}{\delta x} = \frac{\delta ...
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0answers
72 views

Charge Density Triple Integral to infinity

OK, I'm given a charge density function $\displaystyle \frac{2\cdot10^{-4}}{1 + \rho^3}$, and the spherical coordinate of $0 \le \rho < \infty$. Task: find total charge of cloud. Is the answer ...
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2answers
183 views

Newton's Method for Roots of Polynomials

The standard way to use Newton's Method for finding a root of a polynomial $p(x)$ is to use the iteration formula $$x_{n+1}=x_n-{p(x)\over p'(x)}$$ I recently thought of a new way of finding the ...
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3answers
141 views

Use Lagrange Multipliers to find the absolute extrema

Use Lagrange Multipliers to find the absolute extrema (if any) of: $f(x,y) = 4x^2 + 9y^2$; subject to $2x +3y = 6$. Using Lagrange I end up with one point: $(\frac{3}{2}, 1)$ I'm just not sure how ...
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1answer
78 views

Wave Equation Descending from 2D to 1D

I am stuck deriving the familiar d'Alembert formula using the Method of Descent, going from 2D to 1D. After using Kirchhoff's Formula, writing the solution independently of $y$ and some manipulations, ...
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2answers
61 views

Solving problems using the *definition* of differentiability

There is a problem in my textbook, that I could not solve and was not able to understand the solution to. The problem had part a, b, c, d. Only a were solved. I am out of luck. I hope, if somebody ...
2
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1answer
51 views

Double integral and polar coordinates

Please, help me solve this double integral $$\int^{2\pi}_0d\varphi\int^{2}_1\frac{1}{\sqrt{\rho^3\cos^3\varphi+\rho^3\sin^3\varphi}}\rho\,d\rho$$ I really don't know how to figure out and carry of ...
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2answers
46 views

Is it possible to find a function if we know its differential?

Not something we were taught at uni yet, just something that peaked my curiosity. If I was given a derivative of a scalar function, for example $f'(x)=x$ then I know that $f(x)=\frac{x^2}{2}$ (let's ...
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1answer
21 views

double integration via u-subtitution

I'm having trouble with this double integral, maybe someone can help me out: $\int_1^2 \int_0^{lnx} 4x \ dy dx$ My attempt: $$\int_0^{lnx} 4x \ dy = 4xy \big |_{y= 0}^{y= lnx} = 4x \ln(x) $$ $$ ...
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1answer
45 views

Basic surface integral with Stokes.

Calculate surface integral $\iint_S \nabla \times \mathbf{F} \bullet \mathbf{n} \; dS$ with stokes, when $$\mathbf{F}=\left\langle\frac{5y(z-1)}{6},xz, 6e^{xy}\cos{z}\right\rangle$$ and $S$ is surface ...
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1answer
46 views

Question with divergence theorem

Calculate flux of field $$\mathbf{F}=(3x^2 y+6)\mathbf{i}+\left(\frac{x y^2 +1}{3}\right)\mathbf{j}+(3yz^2+3)\mathbf{k}$$ in a box where it's opposite angles are in $(1,1,1)$ and $(2,2,2)$ and it's ...
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3answers
84 views

Optimize function on $x^2 + y^2 + z^2 \leq 1$

Optimize $f(x,y,z) = xyz + xy$ on $\mathbb{D} = \{ (x,y,z) \in \mathbb{R^3} : x,y,z \geq 0 \wedge x^2 + y^2 + z^2 \leq 1 \}$. The equation $\nabla f(x,y,z) = (0,0,0)$ yields $x = 0, y = 0, z \geq 0 $ ...
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2answers
30 views

Planes and surfaces and normal vectors?

Is a plane the same thing as a surface? and is the normal vector the same at every point on both??
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0answers
80 views

Understanding double Riemann sums

I have the following two parts of a question, and I merely want to understand what is being asked of me: 1) Divide the region of integration into 4 rectangles of equal width and height. By ...
2
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1answer
133 views

Integration w/ Change of Variables

folks. I've got this question: Let $D$ be the region $\{(x,y) ~|~ 0 \leq y \leq x, 0 \leq x \leq 1\}$. Evaluate: $$\iint_D (x + y) dxdy$$ by making the change of variables $x = u + v$, $y = u ...
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1answer
450 views

Use Lagrange Multipliers to show the distance from a point to a plane

I'm trying to use Lagrange multipliers to show that the distance from the point (2,0,-1) to the plane $3x-2y+8z-1=0$ is $\frac{3}{\sqrt{77}}$. Our professor gave us two hints: We want to minimize a ...
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1answer
29 views

Evaluate $\iint_s\text{curl}\textbf F\cdot \textbf {n}dS$

Let $\textbf F=<xy,yz,zx>$ and $S$ be the upper half of the ellipsoid $\displaystyle \frac {x^2}{4}+\frac {y^2}{9}+z^2=1$. Evaluate $\iint_s\text {curl}\textbf F\cdot \textbf {n}dS$ I know the ...
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1answer
33 views

Let $S$ be given by $\vec r(u,v)=\langle u\cos v,u\sin v,v\rangle$. Find the tangent plane to the surface at $\vec r(1,\frac {\pi}{4})$.

Let $S$ be given by the vector valued function $\vec r(u,v)=\langle u\cos v,u\sin v,v\rangle$. Find the tangent plane to the surface at $\vec r\left(1,\frac {\pi}{4}\right)$. What I did: $$\vec ...
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0answers
27 views

How to get 3d direction vector of a 2d function?

If I have a function $f(a,b)$, where I can plot it on a 3d graph, using $a$ and $b$ as $x,y$ coordinates and $f(a,b)$ as the $z$ coordinate (i.e. ...
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2answers
36 views

higher partial derivative

I'm confused here: $$f(x,y) = \sqrt{x^2 + y + 4}$$ I got: $$\frac{\partial f}{\partial x} = x(x^2 + y + 4)^{-\frac{1}{2}}$$ $$\frac{\partial f}{\partial y} = \frac{1}{2}(x^2 + y + ...
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1answer
112 views

Fundamental vector product calculation

Given the surface $S\in \mathbb{R}^3$; $y=x^2+z^2$ compute the fundamental outward normal. I seem to get two different answers depending on my parametrisation. Using ...
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1answer
75 views

Find maximum on elipsoid using implicit function theorem…again

I feel like im drowning this site with question about implicit function theorem but I really do not understand how I can find the differential. we are given elipsoid $x^2+y^2+z^2+xy+yz-54=0$ We are ...
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1answer
41 views

How to determine the boundary curve of a surface? [StewartP1097 16.8.15]

$1.$ In the main, how do you determine the boundary curve? I referenced how to find the boundary curve of a surface, but the answer thereunder refers to the specific example of a Mobius strip. ...
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2answers
238 views

Without Stokes's Theorem - Calculate $\iint_S \operatorname{curl} \mathbf{F} \cdot\; d\mathbf{S}$ for $\mathbf{F} = yz^2\mathbf{i}$ - 2013 10C

2013 10C. Consider the bounded surface S that is the union of $x^2 + y^2 = 4$ for $−2 \le z \le 2$ and $(4 − z)^2 = x^2 + y^2 $ for $2 \le z \le 4.$ Sketch the surface. Use suitable ...
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0answers
31 views

Implicit Function Theorem, local form

I am currently trying to understand the proof of the implicit function theorem in the local case, i.e. where $F(x,y)=0$. Consider $F(x,y)=x^2+y^2-1$ and let $(x_0,y_0)=(0,1)$ so that $F(x_0,y_0)=0$ ...
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1answer
20 views

Evaluate an integral over $\mathbb{R}^{3}$ and Green's Theorem

Let $p(x_{1}, x_{2}, x_{3})$ be a smooth function in $\mathbb{R}^{3}$ decaying sufficiently rapid as $|x| \rightarrow \infty$. Why is $$\int_{\mathbb{R}^{3}}p_{x_{i}}\, dx = 0?$$ By the Gauss-Green ...
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0answers
43 views

Validity of following solution

I would like to know if my following solution to a problem I previously posted is valid or not. Problem: If $S$ is a sphere and $F$ satisfies the hypotheses of Stokes' theorem, show that ...
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1answer
97 views

Surface integrals and Jacobian

When do we need to use the Jacobian for surface integrals? I'm asking because when you parametrize a surface $S\in \mathbb{R}^3$ then surely there is no need to change coordinates (essentially ...
0
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1answer
27 views

Find the following partial derivatives?

$$F(x,y,z)=x^8y^2+\sin(y^3z^2)+3=0$$ Find $∂z\over∂x$ and $∂z\over∂y$. I'm pretty confused since I'm only used to finding partial derivatives of something like $∂F\over∂x$ or $∂F\over∂y$. Any help ...
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2answers
512 views

Why does the following limit not exist?

The limit is $$\lim\limits_{(x,y)\to(0,0)}\frac{\sqrt y}{\sqrt x}$$ I thought it was 1. Also what about $$\lim\limits_{(x,y)\to(0,0)}\frac{ y}{ x}$$
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1answer
37 views

Derivative of a function of a matrix $f(B) = x^T(AB)^ky$

I have a function of the form $f(B) = x^T(AB)^ky$ where $x$ and $y$ are column matrices, $A$ and $B$ are square matrices, and $B$ is a diagonal matrix, and $k$ is an integer constant. I want to find ...
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1answer
39 views

Validity of homework question?

Is the following homework problem posed correctly? If $S$ is a sphere and $F$ satisfies the hypotheses of Stokes' theorem, show that $$\iint_{S}F\cdot ds =0.$$ My intuition tells me that the problem ...
0
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1answer
50 views

Chain rule application to find second partials

$x = 2r-s$, $y= r + 2s$, $V = f(x,y)$. Find $\frac {\partial^2 V}{\partial y \partial x}$ in terms of derivatives of $V$ with respect to $r$ and $s$. My work so far: $$\frac {\partial x}{\partial r} ...
0
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1answer
81 views

Differential Equations - Method of Undetermined Coefficients for products of polynomials and sines

Consider $y''+y= 2x \sin (x)$ I have the solution for the homogeneous equation. Now i am trying to guess a particular solution for: $2x \sin (x)$ My first guess was: $(Ax+B) \cos x + (Cx +D) \sin ...
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1answer
39 views

Partial derivatives + Taylor's Formula in several variables

Given a function $f(x) = (x_1+...+x_n)^k$, how do we show that $$D_1^{j_1}\cdots D_n^{j_n}f(x) = k!$$ if $j_1+...+j_n = k$?
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1answer
703 views

Multivariable calculus - find derivative using implicit differentiation

Short simple question which i managed to solve partially. we are given the equation $x^2+y^2-z^2+xz-yz-1=0$. Show using the implicit function theorem that this equation sets in the neighborhood of ...
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1answer
33 views

Elements of a Negative Semidefinite Matrix

Use the definition of a negative definite matrix to show that if A is negative semi-definite: $$A_{ii} ≤ 0 \ \forall i $$ I know the definition (in terms of quadratic form) and the equivalent rules ...
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1answer
34 views

Transform $f''_{xy}$ into $u$ and $v$.

Transform $f''_{xy}$ into $u$ and $v$. See paper below. I miss out on one term, I suspect that I forgot the product rule somwhere, but I cannot tell where.
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0answers
36 views

Treating partials as regular fractions

$\frac{ \partial ^2 f}{ \partial x ^2} = \frac{\partial f}{ \partial x} \cdot \frac{\partial f}{ \partial x} $ I saw my teacher just multiply two partials, to get the second derivative, se image. ...
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1answer
56 views

Volume of the Region bounded by $y = 2x^2 +2z^2$ and the plane $y=8$

I have the find the volume of the region bounded by the paraboloid $y = 2x^2 +2z^2$ and the plane $y=8$. Is the volume (using triple integrals) just ...
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1answer
69 views

double area integrals over coherence functions on circles

I am having trouble showing the following, which shows up from coherence theory: $\frac{\pi b^2}{\alpha^2}(1-J_0^2(\alpha b)-J_1^2(\alpha b))=\int_0^{2\pi}\int_0^b\int_0^b r_1r_2\frac{J_1\left ...
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1answer
54 views

Fundamental Theorem of Calculus on line integrals

I've been shown by my lecturer how to find a function $f$ given $\nabla f$ using the FTC for line integrals. You use $$\int_C {\nabla f}\cdot d\textbf{r}=f(\textbf{x})-f(\textbf{x}_0) $$ And consider ...
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2answers
24 views

Description of the boundary of a ball

Solving an optimization problem in multiple variables, I had to examine a function $$ f(x,y,z)=x^2+2yz $$ defined on a ball $$ \{ (x,y,z)\enspace|\enspace x^2+y^2+z^2\leq1 \}. $$ The boundary is then ...
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3answers
321 views

Polar coordinates for $xz$-plane: $z = r\sin\theta$ ? [Stewart P1091 16.7.25]

$1.$ The unit disk is projected onto the xz-plane, so shouldn’t $x = 1\cos \theta$ and $\color{red}{z = 1 \sin \theta} $? User Semsem below kindly identified the problem: The normal to the ...
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2answers
122 views

$\theta$ for Triple Integral above paraboloid $z = x^2 + y^2$ and below $z = 2y$ [Stewart P1011 15.8.37]

$\bf\sf37.$ Evaluate $\iiint_E z\,dV,$ where $E$ lies above the paraboloid $z=x^2+y^2$ and below the plane $z=2y.$ In cylindrical coordinates the paraboloid is given by $z=r^2$ and the plane by ...
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1answer
37 views

Proving positive-definiteness using the characteristic equation

How does one prove that $q(x,y,z) = 2x^2+5y^2+2z^2+2xz$ is positive definite by solving its characteristic equation?
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2answers
51 views

Third degree Taylor polynomial in two variables

How does one find the third-degree Taylor polynomial of $f(x,y) = (x+y)^3$ at the points $(0,0)$ and $(1,1)$? Many thanks
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1answer
29 views

Accumulation of zeros for a $C^3$ function

Is there a $C^3$ function $v(x,y)$ such that : $v\left(\frac{1}{n},0\right)=0$ for all $n\in\{1,2,\ldots\}$ $v\left(\frac{1}{n+\frac{1}{2}},0\right)\neq0$ for all $n\in\{1,2,\ldots\}$ ...
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0answers
31 views

Optimization non-compact region

I've unsuccessfully been looking all over the web for examples on optimizing multivariable, real-valued functions over non-compact regions. As I've understood it, such optimizations are essentially ...
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0answers
385 views

Volume enclosed by two spheres (triple integral, cylindrical coordinates)

The question: Find the volume of the solid enclosed by the sphere $x^2 + y^2 + z^2 - 6z = 0$ , and the hemisphere $x^2 + y^2 + z^2 = 49 , z ≥ 0$ I set up the triple integral ...