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
37 views

continuity of a function from the plane to the plane [closed]

Let $: R^2 \to R^2 $ be $$ f(x,y) = \begin{cases} \frac{x^2-y^2}{x^2+y^2}, & \text{if }(x,y)\text{ =(0,0)} \\ 0, & \text{if }(x,y)\text{ $\neq (0,0)$} \end{cases} $$ My attempt: Since $f$ ...
1
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3answers
58 views

Question regarding multivariable chain rule…

Suppose that $f : \mathbb R^2 \to \mathbb R$ is some function, and $g :\mathbb R^2 \to\mathbb R$ is defined by $g(x, y) = f(f(x, y), x)$. Write $d(g(x, y))/dx$ and $d(g(x, y))/dy$ in terms of partials ...
1
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1answer
57 views

Multivariable differential equation

Given $u=f(2x-y)+g(x-2y)$, show that $$2 \frac{\mathrm{d}^2 u}{\mathrm{d}x^2} + 5 \frac{\mathrm{d}^2 u}{\mathrm{d}x\,\mathrm{d}y} + 2\frac{\mathrm{d}^2 u}{\mathrm{d}y^2} = 0.$$ I'm not even sure where ...
1
vote
1answer
37 views

Variational characterization of gradient?

Let $f: \mathbb{R}^n \to \mathbb{R}$ be a differentiable function. One way to define the gradient of $f$ is as the vector whose inner product with any other vector gives the directional derivative in ...
3
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2answers
152 views

On the hessian matrix and relative minima

I'm asked to prove the following statement: Let $\mathbb{A} \subseteq \mathbb{R}^n$ an open subset and $f: \mathbb{A} \to \mathbb{R} / f \in \mathbb{C}^3 \wedge (P \in \mathbb{A} : \nabla f(P)=0)$. ...
10
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5answers
282 views

Is line element mathematically rigorous?

I know differentials (in a way of standard analysis) are not very rigorous in mathematics, there are a lot of amazing answers here on the topic. But what about line element? $$ds^2 = dx^2 + dy^2 ...
0
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1answer
83 views

Velocity of a particle in polar coordinates

The equations $r = 3\sin(2\theta)$ and $\frac{d\theta}{dt} = 2$ describe the motion of a particle in polar coordinates. Find the velocity of the particle in terms of the unit vectors $u_r$ and ...
0
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1answer
51 views

implicit function theorem and one to one function

Apply the IFT to show that no $C^1$ function $F:\mathbb{R}^2 \to \mathbb{R}$ can be one to one near any points of its domain. So I know that the theorem says that if we have a point $(a,b)$ such that ...
0
votes
1answer
154 views

volume of a cylindrical shell

Find in cylindrical coordinates the volume of a object that is above the surface $z=\sqrt[4]{x^2 + y^2}$ and inside $x^2 +y^2 + z^2 =2$. We use $$V=\int_0^{2\pi}\int_0^1\int_0^{z^2} 1 ...
0
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1answer
26 views

How do I derive the bivariate Poisson density function

How do I derive the bivariate Poisson density function: ...
1
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2answers
150 views

Line of intersection/quadric surfaces

Let $C$ be the curve of intersection of the cylinder $\frac{x^2}{25}$ + $\frac{y^2}{9}$ = $1$ with the plane $3z = 4y$. Let $L$ be the line tangent to $C$ at the point $(0,-3,-4)$. What is the ...
3
votes
1answer
264 views

Curvature of a curve lying on a sphere?

This is a sample question from a multivariate calculus class. Any insight would be appreciated. Suppose the curve $\mathbf{r} = \mathbf{r}(s)$ is parametrized by a natural parameter and lies on the ...
1
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2answers
84 views

Plane intersection

The planes $5x + 3y + 2z = 0$ and $ 2x + 8y - 5z = 0$ intersect. Find the equation of the intersecting line. I get the parametric equation: $x = t$ y = $\frac{29}{34}t$ z = $\frac{-121}{170}t$ ...
2
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0answers
63 views

Mean Value Theorem help

once again. I wish to prove that $$\frac{2}{\pi} = \cos\left( \frac{\pi t}{2}\right) + \sin\left( \frac{\pi}{2} ( 1-t ) \right)$$ for some t in the interval $( 0, 1 )$ given the function $$f( x, y ...
2
votes
0answers
74 views

Distance between point and plane

Find the distance from the Point $A = (1,0,2)$ to the plane passing through the point $(1,-2,1)$ and perpendicular to the line given by the parametric equations: $$ \begin{align} x & = 7, \\ y ...
1
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0answers
93 views

Calculate partial derivative of multivariable function using the chain rule

Calculate $\frac{\partial u}{\partial t} \text { where } u = x^2 + y^2 + xz \text{ and } x = \sin t, y = e^t, z = t^3.$ The book claims the answer is $\sin2t + 2e^{2t} + 3t^2\sin t + t^3\cos t$. ...
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0answers
33 views

Bounds of integration after a general rotation in 2D?

I have the integral $\int_h^\infty dx \int_d^e dy \ e^{-ax^2+cxy-by^2}$ I want to solve this by using a general coordinate rotation $x= x^\prime cos\theta-y^\prime sin\theta$ $y= x^\prime ...
5
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2answers
137 views

Answer Says $\lim_{(x,y)\rightarrow(0,0)}\frac{x^3y^2}{x^4+y^6} = 0$. I say DNE. What did I do wrong?

I was asked to find $$\lim_{(x,y)\rightarrow(0,0)}\frac{x^3y^2}{x^4+y^6}$$ Observe that setting y=mx results in $$\lim_{(x,mx)\rightarrow(0,0)}\frac{x^3(mx)^2}{x^4+(mx)^6} = 0$$ The textbook ...
0
votes
1answer
99 views

Advanced calculus

Let $Q(x)=\sum_{i,j=1}^{n} c_{ij}x_ix_j >0$ for every $x\neq 0$ where $c_{ij}=c_{ji}$ for $i,j=1,2,\ldots,n.$ Show that $$\int ...
4
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1answer
62 views

Double integrals in polar coordinates

Determine the domain of $D=\{(x,y) \in \Bbb{R}^2 |x\in [-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}],y\in[|x|,\sqrt{1-x^2}]\}$ in polar coordinates and draw it. Also how would you integrate $$\int\int_D ...
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0answers
21 views

Change of variables — from 1 to 2--and setting up the differentials.

Question: If I had an integral of the form $$\int f(E)\, dE$$ and a relationship relating E to two different variables $$E=ax^2+by^2,$$ when I make the substitution of this relationship into the ...
3
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0answers
45 views

Homework: Second derivative of $\langle Ax, x \rangle$

So let $A \in M_{n}$ and define $f: \mathbb{R}^n \to \mathbb{R}$ by $f(x) = \langle Ax, x \rangle $. Find f' and f''. After some work, I found the first derivative to be $f'(x)(v) = \langle Ax, v ...
4
votes
1answer
367 views

Volume of the first octant under a surface

Find the volume of the first octant region under the surface $\sqrt{x}+\sqrt{y}+\sqrt{z}=1$ I think that the integral should be: $$\int_{0}^1\int_{0}^{\left(1-\sqrt ...
0
votes
1answer
56 views

Total derivative of a function $f:\mathbb R^3\to \mathbb R^3$

Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be given by $$ f(x,y,z) = (x^z,y\tan(x^2z),\frac{1}{x+y})$$ and let $c: \mathbb{R}\rightarrow \mathbb{R}^3$ be given by $$c(t) = (\cos(t), \sin^2(\pi ...
0
votes
1answer
74 views

Negative volume in solids of revolution

Find the volume of the solid of revolution created by rotating the area bounded between the curves $y=x^{2}-2, y = 0, y = -1$ around the line $y=-1$. I have set up the volume in two methods: double ...
0
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0answers
22 views

Find $\alpha$ such that $\lim_{(x, y) \to (0, 0)} \frac{|x|^\alpha y}{x^2 + y^2} = 0$. Can I simply move to polar coordinates?

This is the full exercise: Given function: $$f(x, y) = \begin{cases} \frac{|x|^\alpha y}{x^2 + y^2}, & \text{if }(x,y) \ne (0, 0) \\ 0, & \text{if } (x,y) = (0,0) \\ \end{cases}$$ ...
0
votes
1answer
77 views

Find all points at which the direction of steepest ascent of a function is in a given direction.

How do I find all the points at which the direction of steepest ascent of the function $f(x,y)$=$x^2+y^2-2x-4y$ is in the direction $(\sqrt{\frac{1}{2}},\sqrt{\frac{1}{2}}))$
3
votes
2answers
62 views

Multivariable Calculus Help

Prove that if the function z is defined implicitly by $$F( x - az, y -bz ) = 0$$ where F has continuous partial derivatives, then $$a\frac{\partial z}{\partial x}+b\frac{\partial z}{\partial y} = ...
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0answers
27 views

Show $f$ is differentiable and find the derivative at any given point.

$f:U \to \mathbb{R}, (x,y) \mapsto \sqrt{1-x^2-y^2}, \text{ where } U = \{(x,y)|x^2+y^2<1\}$ Okay, basically my strategy was: assume that partials exist then do them out and, Continuous partials ...
1
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2answers
41 views

How can I calculate $ \int_0^t\int_{-\infty}^{+\infty}\frac{1}{\sqrt{4\pi (t-s)}}\exp\left({-\frac{(x-y)^2}{4(t-s)}}\right) y\ dy\ ds $?

I got the following integral $$ \int_0^t\int_{-\infty}^{+\infty}\frac{1}{\sqrt{4\pi (t-s)}}\exp\left({-\frac{(x-y)^2}{4(t-s)}}\right) y\ dy\ ds\tag{*} $$ when I solve the heat equation using the heat ...
2
votes
3answers
85 views

Prove $\lim_{(x,y) \to (0,0)} \frac{x^3y}{x^4+y^2} = 0$

How would you prove the following limit? $$\lim_{(x,y) \to (0,0)} \frac{x^3y}{x^4+y^2} = 0$$ I think the best way is using the squeeze theorem but I can't find left expression. $$0 \le ...
2
votes
0answers
142 views

Proving differentiability

I'm trying to do Spivak's Calculus on Manifold excersise 2-4. It goes as follows: Let $g$ be a continuous real valued function on the unit circle $\{x\in\mathbb{R}^2:||x||=1\}$ such that ...
-1
votes
1answer
56 views

About the non-constant harmonic function

I want to see the steps for the proof of this result: If $u:ℝⁿ→ℝ$ is a non-constant harmonic function, then $u⁻¹(c)$ is unbounded in $ℝⁿ$ for any $c∈ℝ$
0
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1answer
35 views

Could you please explain this to me?

$$(x-c)^2 + y^2 + z^2=3$$ $$x^2+(y-1)^2 + z^2=1$$ Then corresponding normal vectors will be perpendicular , ie when $(x_0,y_0,z_0)$ is on the intersection, then (using the gradient) ...
1
vote
1answer
54 views

Does a line integral depend continiously on the curve?

Let $\gamma_n: [0,1] \to \mathbb R^2$ be a family of curves which converges uniformly to the curve $\gamma$. Does the line integral $\int_{\gamma_n} \vec{F} \cdot \vec{d}s$ over an arbitrary vector ...
6
votes
2answers
125 views

Relation between exterior (second) derivative $d^2=0$ and second derivative in multi-variable calculus.

What does an exterior (second) derivative such as in $d^2=0$ have to do with second derivatives as in single- or multi-variable calculus? Is this a correct start: Calculus derivatives are good for ...
0
votes
1answer
27 views

Calculate $\int\int_D xy(x^2 + y^2)dx\,dx$ where D is the set that bounded by $1 \le xy \le 2 \text{ and } 5 \le x^2 - y^2 \le 9$

An exercise from the book that I have a difficults to solve.. Calculate $\int\int_D xy(x^2 + y^2)dx\,dx$ where D is the set that bounded by $1 \le xy \le 2 \text{ and } 5 \le x^2 - y^2 \le 9$ ...
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votes
2answers
511 views

Partial derivatives must exist and be continuous on all defined points of $f$ for $f$ to be differentiable?

Today my professor explained that $f(x,y)=\frac{2xy}{(x^2+y^2)^2}$ is differentiable even though $(x,y)=(0,0)$ is not defined. The partials are $\frac{\partial f}{\partial ...
0
votes
2answers
20 views

Distribution Property of Dot Product

Since the dot product has the property that for three vectors $a,b,c$ $a \cdot (b+c) = a \cdot b + a \cdot c$ Is that also true for $(a+b) \cdot (a+b) = a\cdot a + 2a\cdot b + b\cdot b$ ? Thank ...
0
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1answer
36 views

Lagrangian Multiplier Question

I can do question 2 easily but I'm running into some problems proving 1 rigorously. No idea how to go about doing it at all.
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2answers
49 views

Finding work via Line Integrals

The position of an object with mass $m$ at time is $r(t) = at^2 \vec{i} + bt^3 \vec{j}$, where $0 \leq t \leq 1$. Part a asks for the force, which I found to be $2ma \vec{i} + 6mbt \vec{j}$, which is ...
0
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1answer
29 views

Can you give a general expression for the rate of rising water poured in an object?

Given an 3-d object defined by a set of parametric equations (x, y, z), can you write a formula expressing the rate that a liquid rises as it's poured into this object at a constant flow rate? Assume ...
1
vote
1answer
34 views

Can you add potentials if charge redistributes?

Let say we have charged conductor $M$ and we know its potential energy function $V_m(r)$ when $M$ is isolated from any charges. We also have charged conductor $N$ with potential energy function ...
1
vote
0answers
16 views

How do I find the maximum and minimum values of xy on an off-center ellipse?

What's the maximum and minimum of f(x,y)=xy with the constraint $$\dfrac{(x-x_o)^2 }{A^2} + \dfrac{(y-y_o)^2 }{B^2} = 1$$ Using lagrangian multipliers is simple when the center of the ellipse is ...
1
vote
1answer
38 views

Differential Geometry-curves

Let $c:[0,2] \to\Bbb R^3$ be the curve given by $$c(t)=(\frac {t^3}{3},t^2,2t).$$ Then there exists an $m>0$ and a $C^{\infty}$ bijection $f:[0,m]\to [0,2]$ such that $f'(s)> 0$ for every ...
1
vote
2answers
136 views

Volume above cone and below paraboloid.

I need to find the volume above the cone $z=\sqrt{x^2+y^2}$ and below the paraboloid $z=2-x^2-y^2$. I thought about using spherical coordinates and finding $p$, which would be (before simplification) ...
0
votes
3answers
134 views

Calculate $\iint_D x\ln(xy) dx\,dy \text{ where } x = 1, x = e, y = \frac{2}{x}, y = \frac{1}{x}$

Ok this is a sample exercise from the book that I don't know how to solve. Calculate $\iint_D x\ln(xy) dx\,dy \text{ where } 1 \le x \le e , \frac{2}{x} \le y \le \frac{1}{x}$ The answer is ...
2
votes
1answer
49 views

Second Order Derivative of a function $f:R^2\to R^2$

The Exercise: My Work: Part 1: $$ Df=\left( \begin{array}{ccc} D_1f_1 & D_2f_1\\ D_1f_2 & D_2f_2 \\\end{array} \right) $$ $$f_1(x,y)=\sin x+\sin y$$ $$f_2(x,y)=\cos x+\cos y$$ $$ ...
1
vote
1answer
83 views

Integration by parts with complex numbers

Suppose $u$ is a complex-valued wave function $u(x,y,z,t)$. Also, suppose you have the integral $\int(u\overline{u}_{t}+u_{t}\overline{u})dx$. I need to get ...
0
votes
1answer
151 views

Find the average value of the function $F(x,y,z) = x^2y^2z^2$ along the curve.

How do I find the average value of the function $F(x,y,z) = x^2y^2z^2$ along the curve given by $$s(t) = \left( t,\frac{\sqrt{2}}{2}t^2,\frac{t^3}{3} \right).$$ I understand that you need to find ...