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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-1
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1answer
18 views

Prove the following lemma

If $f$ is differentiable at $X_o$, then $f(X)-f(X_o)= (d_{x_o} f)(X-X_o)+ E(X)|X-X_o|$, where E is defined in the neighborhood of $X_o$ and $\lim_{X\to X_o}$ $E(X)=E(X_o)=0$ I don't know how to ...
0
votes
1answer
20 views

What is the derivative of $|Du|$?

Let $u:\mathbb{R}^n\to \mathbb{R}$. Suppose $u$ is differentiable. What is the derivative of $|Du|$? Is it equal to $\text{div}(Du)$? Here div means the divergence. Thank you very much.
1
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2answers
43 views

Prove that the limit of the following function of two variables is zero

I need to prove the following: $$\lim_{(x,y)\to (1 ,2)} \frac{x^2+2xy-6x-2y+5}{\sqrt{(x-1)^2+(y-2)^2}}=0$$ I've tried to solve it by substituting $y=mx$ but I can't get the solution that way. ...
0
votes
2answers
42 views

How to argue that $f(x,y,z)=-x^2-xy-y^2+4yz-8z^2+2xz$ has a global maximum?

Let $f(x,y,z)=-x^2-xy-y^2+4yz-8z^2+2xz$. I know $f$ has a local maximum at $(0,0,0)$ but how do I argue that this is also the global maximum. The solution provided simply states it is a global ...
2
votes
0answers
42 views

Multiple Integral Puzzle [duplicate]

$$I=\int_1^2\int_1^2\int_1^2\int_1^2 {{x_1+x_2+x_3-x_4}\over{x_1+x_2+x_3+x_4}} \,dx_1\,dx_2\,dx_3\,dx_4$$ $I=1/2$ or $1/3$ or $1/4$ or $1$ ? [from ISI-Kolkatta Sample papers] I know that there ...
0
votes
1answer
12 views

Find all real values of $k$ for which the given integral converges.

In the following, $R_k$ is the region $1 \leq x \leq \infty$, $0 \leq y \leq x^k$. $b$ is just a given real number. Find all real values of $k$ for which the given integral converges: \begin{align*} ...
0
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0answers
21 views

Corollary of the inverse function theorem

Let $U\subset \mathbb{R}^{n}$ and $ f:U\to \mathbb{R}^{n}$ injective and class $C^{1}$ such that $\det f'(x)\not=0$ for all $x \in U$. Show that $f(U)$ is open and $f^{-1}:f(U)\to U$ is ...
0
votes
1answer
36 views

What does it mean to “admit” something in vector calculus?

Trying to understand the Helmholtz decomposition has lead me to the concept of a vector potential. From Wikipedia [1]: If a vector field v admits a vector potential A, then [...] I've searched ...
1
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0answers
36 views

$\lim(x^2+y^2)/(x-y)$ when $(x,y)\to(0,0)$ [duplicate]

How can I prove (without polar coordinates) that the $\lim \frac{x^2+y^2}{x-y}$ when $(x,y)\rightarrow (0,0)$ does not exist? Moreover, can I use $(x^2+y^2)=(x+y)(x-y)$ in this case? If I do it seems ...
1
vote
0answers
14 views

Finding the Local Maximum and Minimum?

I am given $f(x,y) = 6y - y^3 - 4x^2y$ and asked to find the critical points. After I have found those I'm asked to find D= $f_{xx}f_{yy} - f_{xy}^2$ and to decide whether I have a local max, min, or ...
1
vote
1answer
33 views

What is the $k$th differential of a composite map?

Let $V$, $W$, and $X$ be normed linear spaces, and let $A$ be an open subset of $V$. Suppose $F \in C^k(A,W)$ and $G \in C^k(W,X)$. What is $d^k(G\circ F)_\alpha$ (the $k$th differential of $G \circ ...
2
votes
0answers
26 views

difficult example of a not differentiable function $f: \mathbb R^2 \to \mathbb R^2$ [on hold]

Give an example of a function $f: \mathbb R^2 \to \mathbb R^2$ so that: 1) all its directional derivatives exist at $(0,0)$ ($D_{\vec u}f(0,0)$ exist for all $\vec u \in \mathbb R^2$ unitary), 2) ...
0
votes
0answers
12 views

show the surface is smooth r(u,v)=((2+sinv)cosu, (2+sinv)sinu, u+cosv)

given the function r(u, v) = [(2 + sin v) cos u, (2 + sin v) sin u, u + cos v] need to show that the surface is a smooth surface according to the regularity conditions, if $\partial_u$r $\times$ ...
0
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0answers
19 views

Suppose that $L$ is a linear transformation of the plane onto itself. Let $u:\mathbb{R^2}\to\mathbb{R}$.

If $v$ is a new function given by $v(x)=u(Lx)$, write down $\partial_{{x_i}{x_j}}u$. Okay so I know you have to use the chain rule and I know L is given by a a 2 by 2 matrix $l_{ij}$ But I don't ...
1
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0answers
25 views

I need help with divergence and gradient?

$$A_z = \mu{\frac{e^{-jBr}}{4\pi r}}∫I(z')e^{jBz'\cos\theta}dz'$$ Midway into my question, I want to compute: $$-j\left( \frac{\nabla(\nabla\cdot A) }{w\mu\varepsilon} \right).$$ Symbols like $ w, ...
0
votes
1answer
36 views

Prove a consequence of the multivariable version of the inverse function theorem

The exercise is the following: Let $f:\mathbb{R}^{n} \to \mathbb{R}^{n}$ that is class $C^{1}$ such that there exists $c >0$ such that $$|f(x) - f(y)| \ge c|x-y|$$ for all $x,y \in ...
0
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0answers
12 views

Finding a Gradient Vector

(1) Consider the function $f(x,y)$. If you start at the point $(2,5)$ and move toward the point $(5,1)$, the directional derivative is 1. If you start at the point $(2,5)$ and move toward the point ...
0
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0answers
56 views

If $f_{xy}$ , $f_{yx}$ are continuous at $(x_{0},y_{0})$,then $f_{x},f_{y}$ are continuous at $(x_{0},y_{0})$?

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ The second edition Let $f$ be a function of two variables,let$(x_{0},y_{0})$ be a point and let $U$ be an open disk with center $(x_{0},y_{0})$.Assume ...
1
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2answers
39 views

Wind chart for vector calculus

If a particle starts its motion at the origin of a Cartesian coordinate system under the influence of a force $F(t) = (1, x(t))$ where $t \ge 0$, find an equation of the path it follows. This seems ...
0
votes
3answers
41 views

Why is $f: \mathbb R^2 \rightarrow \mathbb R$ not differentiable at $0 \in \mathbb R^2$ given limit as $t$ approaches $0$?

Consider $f: \mathbb R^2 \rightarrow \mathbb R$ given by $$f(x_1, x_2) = \frac {x_1^3 -x_2^4} {x_1^2 + x_2^2}$$ if $(x_1,x_2) \neq (0,0)$ and $$f(x_1,x_2) = 0$$ if $(x_1,x_2) \neq (0,0)$. I've proven ...
-1
votes
0answers
17 views

Partial Differential equation of one dimesional heat equation

My Problem is : Show that $u(x,t)=\int_{0}^{\frac{x}{2\sqrt{t}}}e^{-s^2}ds$ satisfies 1-d heat equations $\frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2}$. Sketch the level curve of ...
1
vote
1answer
27 views

For every $h \in \mathbb R^n$ the limit $\lim_{t \rightarrow 0} \frac 1 t (f(x_0 + th)-f(x_0))$ exists and is equal to $D_{f_{x_0}}(h)$?

Suppose $U \subset \mathbb R^n$ is open and $f: U \rightarrow \mathbb R^m$ is differentiable in $x_0 \in U$. I want to show that for every $h \in \mathbb R^n$ the limit $\lim_{t \rightarrow 0} \frac ...
1
vote
1answer
19 views

Determine $\frac{dz}{dt}$ at the point where (s,t) = (2,1)?

I am solving this problem for Lon Capa. It is an online website where they give me questions and I have to solve the answer and then type it in. Sometimes my answers are correct and I'm just typing it ...
0
votes
1answer
70 views

A little question about Clairaut's Theorem

Clairaut's Theorem Let $f$ be a function of two variables,let$(x_{0},y_{0})$ be a point and let $U$ be an open disk with center $(x_{0},y_{0})$.Assume that $f$ is defined on $U$ and its partial ...
0
votes
1answer
20 views

Show that $F$ is not a one-to-one transformation

Given $$F(x,y)=(x-y,y^2-x-2)=(u,v),$$ how to show that this transformation is not one-to-one? And at which points $F$ is locally one to one? While I was drawing this transformation I found that ...
1
vote
1answer
16 views

Is the integrand of this integral singular?

In a book I'm reading, integrals like the following appear and the author says the integrand is not singular (or perhaps, integrable) around the origin. $$ ...
7
votes
1answer
77 views

Is it ever easier to show differentiability than continuity?

I'm TAing a course right now in multivariable calculus and in the lecture notes the professor gave the students the theorem stating that differentiability implies continuity, as well as another ...
0
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0answers
17 views

Euler-Lagrange Equation and “Eigen Value ”

The Eigen value $\lambda(t)$ which is characterised by the Rayleigh quotient (where $t$ is a scalar variable): $$R(u,\Omega_t)= \frac{\int_{\Omega_t} |\nabla u|^2 dy }{\int_{\Omega_t} u^2 dy}$$ ...
0
votes
1answer
29 views

Problem in Gradient operator and Kronecker delta function

I have this expression $$\nabla_{i}\nabla_{j}\Big(\frac{1}{r}\Big)$$ Where $r$ is a distance. I tried this, but encountering manipulations of $\delta_{ij}$ with $\hat{r_i},\hat{r_{j}}$ and still ...
0
votes
1answer
20 views

Volume of Revolution $f(x) = x^2$

Suppose you are given $y = f(x)$ I want to use double integrals, instead of the traditional washers. Suppose even better, $f(x) = x^2$ Find the volume of $f(x) = x^2$, $x = 0$, $x = 4$, $y = 0$ ...
0
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0answers
12 views

Is there a difference between inequality and equality sign when using Lagrange multiplier?

For example, find the extreme values of z=xy subject to the condition x+y=1 This is quite simple example of finding extreme using Lagrange multiplier When the constrain is changed from x+y=1 to ...
0
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0answers
20 views

Partial Derivative of Function

I can't follow a step in one of my textbooks... It says that given a function $f(x-vt)$, we know that $\frac{\partial f}{\partial x} = -\frac{1}{v} \frac{\partial f}{\partial t}$. I think this comes ...
0
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0answers
13 views

Differentiability question (function of two or more variables)

Imagine I have a function $f:\mathbb{R}^m\times \mathbb{R}^n \rightarrow \mathbb{R}^d$, denote by $|\cdot|$ the Euclidean norm in all spaces and I assume the following hypotheses: there is a finite ...
0
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0answers
22 views

Problem about Gradient Vectors

You are at the point corresponding to $(2,3)$ in a valley whose elevation at a point $(x,y)$ is given by the function $$h(x,y)=\frac{(x+y)}{(1+x^2)}$$ (Give your answers as vector for the following ...
0
votes
2answers
33 views

Partial derivative problem on absolute value function

Find the first and second order partial derivatives of $f(x,y)=|2x^2-y|$. I start with limit definition but not able to solve. please help.
0
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0answers
27 views

Leibniz Rule in Improper Integrals?

http://www.math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf In the above link, you will find a proof and various examples of the Leibniz rule. The rule given applies to integrals with finite ...
0
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0answers
10 views

Nontrivial chain rule diagrams, how to write chain rule for them and is there implications or constraints in f that arises in the process?

(This is a short version of the question intended to post, because the original is just TOO LONG) From this link and various others (including 2 links deleted by the system as part of the clean up ...
2
votes
0answers
16 views

problem book on Functions of Two Real Variables

please refer a problem book on Functions of Two Real Variables: limit, continuity, partial derivatives, differentiability, maxima and minima. Method of Lagrange multipliers, Homogeneous functions ...
0
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0answers
21 views

Minimize $f(X)=trace\left(X^{T}\begin{bmatrix} 1&0\\0&16\\ \end{bmatrix}\right)$

Minimize $$f(X)=trace{\left(X^{T}\begin{bmatrix} 1&0\\0&16\\ \end{bmatrix}X\right)}$$ subject to the condition $g(X)=det(X)=1$. Then for taking $X=\begin{bmatrix} ...
0
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0answers
12 views

Show that $f(\vec{x})=\vec{c}$ has atleast one solution for $||c||$ sufficiently small

Let $f: \mathbb{R^{k+n}} \to \mathbb{R^{n}}$ be ${C^1}$. If $$Df_{\vec{a}}$$ has rank $n$, then Show that $f(\vec{x})=\vec{c}$ has atleast one solution for $||c||$ sufficiently small. My try: I ...
0
votes
1answer
25 views

Interchange limit of one variable with partial derivative of another variable

Consider a function $f(t,y)$ where $t,y \in \mathbb{R}$. It is given that $ f(t,y)$ is continuous but $\dfrac{\partial f(t,y)}{\partial t}$ suffers a jump at at $t= x$. (The function $f$ could ...
1
vote
1answer
30 views

Why line integral of f(x.y)=(x.y) is not zero along the circle?

I am asked to determine whether f(x.y)=(x.y) is gradient or not. It is clear that there exists a function g whose derivative with respect to x and y is equal to first and second component of f. ...
0
votes
1answer
17 views

How to find $f_{xy}$ if only $f_y$ and $f_x$ are given?

I am given the partial derivative of a function $f(x,y)$ with respect to $x$ is $4x^2-36$ and the partial derivative with respect to $y$ is $5y+20$. How to find $f_{xy}$ ?
2
votes
1answer
74 views

Showing $f=0$ almost everywhere

Let $\psi_n(x)=e^{-x^2/2}P_n(x)$ where $P_n$ is a degree $n$ polynomial with real coefficients. Assume that $$\int_{\mathbb{R}}e^{-x^2/2}P_n=0.$$ Suppose that for any $f\in L^2$, such that ...
0
votes
0answers
15 views

Question about uniform wire and its application to find centroid

A uniform wire has the shape of that portion of the curve of intersection of the two surfaces x^2+y^2=z^2 and y^2=x connecting the points (0.0.0) and (1.1.square root 2) Find the z-coordinate of its ...
2
votes
0answers
36 views

Bounded integral operator

So if $k:\mathbb{R}^2\rightarrow\mathbb{R}$ such that $$\sup_{x}\int |k(x,y)|dy<\infty \ \ \ \sup_{y}\int|k(x,y)|dx<\infty$$ How do you show in this case, $Kf(x)=\int k(x,y)f(y)dy$ is a bounded ...
-2
votes
0answers
31 views

How do I integrate a function of one variable with respect to another? [on hold]

If $y = f(x)$, how do I find $$\int f(y) dx$$ without knowing what $y$ is in relation $x$.
0
votes
0answers
21 views

Explanation of differentiating implicit functions

$F(x,y,z)=0$, where $F \in C^2$ in some neighborhood of point (a,b,c) in which $F(a,b,c)\neq 0$ For the constant function $(x,y)->F(x,y,f(x,y))$ based on ...
0
votes
1answer
23 views

Finding Critical points of a matrix equation

Consider the function $F: M_{2}(\mathbb{R}) \to \mathbb{R^{2}}$ defined by $F(X)=\left[trX ,detX\right]^{T}$. Show that the critical points form a straight line and critical values form a parabola ...
2
votes
0answers
19 views

Regular parametrised curve and its length

$\newcommand{\Reals}{\mathbf{R}}$Consider the function $r : [1/4, 3/4] → \Reals^2$ defined as $r(t) = (\sqrt{t}, \sqrt{1−t})$. I need to determine whether it is regular and to compute its length. I ...