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).

learn more… | top users | synonyms

1
vote
2answers
57 views

Do projections on $\mathbb{R}^2$ transform straight lines to straight lines?

A linear transformation $P:\mathbb{R}^{2} \longrightarrow \mathbb{R}^{2}$ is called projection if $P \circ P =P$. The question is: If $P$ is a projection then $P$ transforms straight lines ...
0
votes
1answer
28 views

Multivariable Calculus Application Question: Utility and MCRS

If a student has a utility function given by $$U(x_1, x_2) = −x_1 + > 10x_ 2^2 − 2x_1x_2$$ where $X_1 = 5$ and $X_2 = 20$. If this student was to eat $5$ less hot meals per month, estimate the ...
1
vote
1answer
88 views

line integral…

Calculate $$\int_Γ f \, d\ell$$ for $f(x,y) = y, \; y=x^{1/2}$, $ x $ is in $[2,6]$. I know (now) that it means that: $$\int_\Gamma f \, d\ell=\int_a^b f(\Gamma(t)) \cdot \|\dot\Gamma(t)\| \, dt$$ ...
1
vote
1answer
34 views

Find and classify the critical points of the function $f(x,y) = x^3 +2y^3 - 3x^2 -24y + 6$

I have to find and classify the critical points of the function: $$f(x,y) = x^3 +2y^3 - 3x^2 -24y + 6$$ I have said that $$f_x = 3x^2 -6x=0 $$ $$3x(x-2)=0$$ $$x=0, 2$$ $$f_y=6y^2-24=0$$ $$y=±2$$ I ...
2
votes
1answer
81 views

How can I find the point (X, Y, Z) which minimizes this quantity?

I have a number of equally powerful light sources $L_i, 1 \le i \le N$ at points within a cube $(x_i, y_i, z_i), -1 \le x_i, y_i, z_i \le 1$. The intensity of each light falls off with distance ...
0
votes
1answer
111 views

Proof of the rank theorem in Rudin's PMA book

I am studying Rudin's proof of the rank theorem (theorem 9.32 in Principles of Mathematical Analysis.) We have an invertible function $H(x)$ defined on an open set. He claims we can "shrink" the open ...
2
votes
0answers
50 views

The 3rd and 4th Critical Point?

I must find and classify all the critical points in the following function: $$ f(x,y)= x^2 + y^2 +x^2y +4$$ I have said that $$f_x=2x+2xy=0$$ $$ 2x = -2xy$$ $$ \frac{ 2x}{\ -2x}=y $$ $$y=-1$$ $$f_x = ...
0
votes
1answer
54 views

Multivariable Calculus application

A firm is producing cylindrical containers to contain a given volume. Suppose that the top and bottom are made of a material that is $N$ times as expensive as the material used for the side of ...
0
votes
2answers
43 views

Two-dimensional Taylor linearisation

I have to perform a first order taylor expansion of a function $f(\vec{x}) = f(x+u,y+1)$ at the point $\vec{a} =(x,y)$. My solution reads $$ f(\vec{x}) \approx f(x,y) + \left( \begin{matrix} ...
0
votes
3answers
672 views

How to treat Dirac delta function of two variable?

We can treat one variable delta function as $$\delta(f(x)) = \sum_i\frac{1}{|\frac{df}{dx}|_{x=x_i}} \delta(x-x_i).$$ Then how do we treat two variable delta function, such as $\delta(f(x,y))$? ...
4
votes
1answer
109 views

Why generalize the derivative for multivariable functions? [duplicate]

Sorry if this is a dupe (did a search, couldn't find anything). In single variable calculus, if the following limit exists: $$\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h},$$ then this expression ...
4
votes
0answers
114 views

Differential forms and determinants

2-forms are defined as $du^{j} \wedge du^{k}(v,w) = v^{j}w^{k}-v^{k}w^{j} = \begin{vmatrix} du^{j}(v) & du^{j}(w) \\ du^{k}(v) & du^{k}(w) \end{vmatrix}$ But what if I have two concret ...
1
vote
2answers
65 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 ...
0
votes
0answers
24 views

What order should I evaluate divergence and coordinate transformation if I want to use a different coordinate system?

I have a vector field in Cartesian coordinates. I need to find its divergence, but I need the divergence to be in spherical coordinates. However, the divergence of this field is far easier to ...
0
votes
1answer
34 views

Trigonometric Partial Derivative

I need to find $$\frac{\partial Z}{\partial U} \text{ and } \frac{\partial Z}{\partial V}$$ for a $z=f(x,y) = \cos(xy) + y\cos(x)$. After a bit of an internet search, I think I have found the ...
0
votes
1answer
34 views

Partial Derivative Stickler.

I am having trouble with a question with partial derivatives. Here is the question: Let $\rho = \sqrt{x^2 + y^2 + z^2}$ Show that $$\frac{\partial ^2\rho}{\partial x^2} + \frac{\partial ...
0
votes
0answers
41 views

Theorem 4.6 in Spivak's Calculus on Manifolds

Could you elaborate on the proof please? This is how I would prove the theorem: Since $\Lambda^n(V)$ is $1$-dimensional, $\omega=\alpha(\phi_1\wedge\phi_2\wedge...\wedge\phi_n)$ for some $\alpha$ ...
0
votes
2answers
42 views

Having a bit of trouble with min/max distance from sphere to point

The sphere is $x^2 + y^2 + z^2 = 81$ and the point is $(5,6,9)$ I am using Langrane multipliers , but the answers I am getting are so far off. I will post my system of equations soon. I found ...
1
vote
1answer
35 views

Vectors with given angle and magnitude

Give an example of vectors $\mathbf{v}$ and $\mathbf{w}$ such that the angle between $\mathbf{v}$ and $\mathbf{w}$ is $\frac{2\pi}{3}$ and $\|\mathbf{v} \text{ x } \mathbf{w}\|=\sqrt{3}$. Should I ...
0
votes
0answers
50 views

Paraboloid Curvature calculation methods

If we have a paraboloid generated as a surface of revolution of the 2d function $f(x)=ax^2+b$, the equation of the 3d graph is $f(x,y)=ax^2 + ay^2+b$. The gaussian curvature of a 3d graph $f(x,y)$ is ...
3
votes
2answers
125 views

Solving a particular system of differential equations

The problem I'm trying to solve is this: $X'(t) \in \mathbb{R}^3 \,, \, \omega = (\omega_1,\omega_2,\omega_3) $ Find the general solution for $$X'(t) = \omega \times X(t)$$ After doing the cross ...
1
vote
1answer
39 views

Cobb Douglas Difficulty

Show that the Cobb-Douglas production function, for Labour costs L and Capital costs K, $P(L, K) = AL^{\alpha}K^{1-\alpha}$ satisfies the equation: $$L\frac{\partial P}{\partial L} + ...
1
vote
1answer
37 views

Need Help Understanding How To Integrate With An Implicit Variable

My calculus is really rusty (damn Mathematica/Matlab) and I was wondering if anyone could help me with an equation I am having trouble integrating. I have attached a snapshot of the paper I am trying ...
0
votes
0answers
36 views

A question regarding calculus in several variables.

Let $f:\Bbb{R^n}\to \Bbb{R}$ be a continuous function. I refer to this proof of the Mean Value Theorem for several variables. Here $g(t)=f(a+t(b-a))$. Hence ...
2
votes
1answer
46 views

Taking Fourier transform of integral-differential equation

If $u$ is a solution of the equation $$\frac{\partial}{\partial t} u(x,t) + \int_{-\infty}^{\infty} \text{sinc}(x-y) \cdot \frac{\partial^{2}}{\partial y^{2}} u(y,t) \ dy = 0,$$ with initial condition ...
0
votes
1answer
100 views

Transform square region to triangular region

How do you express x and y in terms of u and v so that the region $\{(u,v): 0\le u, v\le 1\}$ is mapped to the triangular region in the $xy$-plane with vertices $(0,0)$, $(1,0)$, and $(0,1)$? Now, ...
6
votes
0answers
51 views

Reference for the fact that a smooth function analytic on every line is itself analytic

Let $f \in \mathcal C^\infty(\mathbb R^p)$ ($p \geq 2$) be a smooth function such that the functions $g_d(t) := f(td)$ are all analytic for all $t \in \mathbb R$ and all $d \in \mathbb R^p.$ (i.e. $f$ ...
2
votes
1answer
116 views

Explanation of Lagrange Equation with Chain Rule

I am just reading through some lecture notes explaining the Lagrange Equation, and I am a bit confused with some chain rule stuff, I get to the part with: $$\frac{\partial F}{\partial y} = ...
0
votes
1answer
34 views

How to calculate the partial derivatives of the composition $F(u(s,t),v(s,t))$?

Could someone help me to understand how to do this problem? I believe Partial Derivatives are used. Thanks!
0
votes
2answers
55 views

Shortest distance between a point an a non linear surface

I am trying to find the shortest distance between a point $P(p_1,p_2,p_3)$ and the surface $z=xy$. Could somebody help me please?
0
votes
1answer
30 views

Rewrite the following surface so that I can graph it.

$z = \dfrac{1+x^2}{1+y^2}$ $ $ I want the part of the surface above the square $|x|+|y|\leq 1$ $ $ OR we can write this square as $-y<x<y$ and $-1<x<-1$ $ $ I have spent hours trying ...
2
votes
2answers
84 views

Double integral where limits are the first quadrant

Evaluate the integral $$\iint\limits_D \frac{1}{(x+y+1)^3} \, dA$$ where $D$ is the first quadrant. In this case, what would the limits of integration be? I'm having trouble moving to polar ...
4
votes
1answer
170 views

Question about path method for multivariable limits

I have to prove that the limit $$\lim\limits_{(x,y) \to (0,0)}\dfrac{x^2}{x+y}$$ does not converge. This is fairly 'easy' to do, but while I was doing it I came across some doubts. I took the limit ...
1
vote
1answer
35 views

Diagonal matrices and integrals

Suppose that $$A=\int_{\alpha}^{\beta} f(B,x)\ dx,$$ where $B$ is a $3\times3$ matrix. The result I'm looking for is that if $B$ is diagonalized with an orthogonal matrix, then is A diagonalized by ...
1
vote
2answers
44 views

Is this proof of multivariable limits legit?

Show that the limit $\lim \limits_{(x,y) \to (0,0)} \frac{x^2 - y^2}{x^2 + y^2}$ does not exists. Step 1) let $x=0$ $\Rightarrow \frac{-y^2}{y^2} =1$ Step 2) let $x = y \Rightarrow \frac{0}{2y^2} =0 ...
1
vote
1answer
47 views

Work done by a force field line integrals

Find the work done by the force field $F(x, y) = \langle 2x \sin(y), 2y \rangle$ on a particle that moves along the parabola $y = x^2$ from $(-1, 1)$ to $(2, 4)$. So to use line integrals to solve ...
1
vote
1answer
77 views

Finding a mistake in the computation of a double integral in polar coordinates

I have to find $P\left(4\left(x-45\right)^2+100\left(y-20\right)^2\leq 2 \right) $ $f(x)$ and $f(y)$ are given, which I will use in my solution below . ...
2
votes
1answer
50 views

Computing double integral

Find $$\iint\limits_D \sqrt{(x-10)^2+y^2}\hspace{1mm}dA$$ where $\{(x, y)\in D \mid x^2+y^2\leq 10^2\}$. I am not sure how to start, every way I have tried so far, ends up into something ugly. All ...
1
vote
1answer
38 views

Finding a function from a vector field

The vector field $F(x, y) = \left(\displaystyle\frac{x}{r^3}, \frac{y}{r^3}\right)$ appears in electrostatics, where $r = \sqrt{x^2 + y^2}$ is the distance to the charge. Find a function $f(x, y)$ ...
3
votes
2answers
270 views

What is the value of this double integral?

Let $C$ be the subset of the plane given by $$ C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | \ 0 \leq x^2 + y^2 \leq 1 \}.$$ Then what is the value of the double integral $$ \int_{C} \int (x^2 + y^2) ...
0
votes
2answers
90 views

How to evaluate this double integral?

Let $C$ be the subset of the plane given by $$C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | -1 \leq x = y \leq 1 \}. $$ Then how to evaluate the double integral $$ \int_C \int (x^2+ y^2) dx dy? $$ My ...
1
vote
2answers
60 views

Verification of the Stokes theorem for the surface that is a part of a cone

Let $S$ consist of the part of the cone $z=(x^2+y^2)^{1/2}$ for $x^2+y^2\leq9$ and suppose $${\bf A}=(-y,x,-xyz).$$ Verify that Stokes theorem is satisfied for this choice of $\bf A$ and $S$. In ...
0
votes
0answers
18 views

Trace of hyperbola $2$ sheets

For $yz$ parallel trace, the book mentions that there is no trace unless $|x|>|a|$, in which case the trace is an ellipse. It is hard for me to visualize this. Any insight would be appreciated, ...
0
votes
2answers
141 views

Laplace transform of noncentral chi-square distribution

I'm interested in non central chi-square distribution. More specifically, i want to derive the laplace transform of noncentral chi-sqruae disribution or density function. Let me know whether it ...
1
vote
1answer
81 views

Bounding the integral of a function by the integral of its derivative

I have no idea where to begin for this question, so any help would be greatly appreciated! Let $\Omega$ be a square with side 1. Show that $$\left(\int_{\Omega} v^2 \, dx \right)^{1/2} \leq \left( ...
1
vote
0answers
64 views

Proving boundedness of a function .

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
3
votes
1answer
63 views

Sketching a surface

If $${\bf F}=2y{\bf i}-z{\bf j}+x^2{\bf k},$$ and $s$ is the surface of the parabolic cylinder $y^2=8x$ in the first octant, bounded by the planes $y=4$ and $z=6$, evaluate $$\int_S{\bf ...
0
votes
1answer
186 views

Finding the area of a triangle from vertices? Linear Algebra

I pretty much did this problem, but I failed to get the few last blanks where they ask the area. Its confusing, they say its half the volume of matrix (u v w) in the start of the question. which means ...
2
votes
3answers
111 views

How do I find a point on the surface of a sphere

How do I find a point on a sphere knowing its radius and center point ? I have a sphere: $$x^2+(y-1)^2+(z+3)^2=16$$ Obviously its center point is $(0,1,-3)$ and its radius is $4$. I am asked to find ...
1
vote
2answers
35 views

Parametrization Question

When computing a line integral, or any integral that requires parametrization, what are you integrating with respect to? For example, if parametrizing in polar coordinates, with $x=r\cos\theta$ and ...