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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2
votes
1answer
45 views

Why is the composition of smooth multivariable functions smooth?

Is an an easy way to see that if $f: U \subset \mathbb R^n \rightarrow V \subset \mathbb R^m$ and $g: V \subset \mathbb R^m \rightarrow \mathbb R^p$ are smooth functions then their composite $g ...
-1
votes
0answers
10 views

Partial derivation of three variables with respect to four variables

I have a set of equations relating variables $[K_1, K_2,K_3,K_4]$ to $[X, Y, Z]$ like: $$K_1=X-C Y cos(Z+ C_1)$$ $$K_2=X-C Y cos(Z+ C_2)$$ $$K_3=X-C Y cos(Z+ C_3)$$ $$K_4=X-C Y cos(Z+ C_4)$$ ...
3
votes
0answers
35 views

Lagrange multipliers in Calculus of Variations

I am trying to learn about Calculus of Variations and I am beginning to see some constrained optimization problems in the domain of functionals, by using Lagrange multipliers. It seems that things ...
-1
votes
1answer
42 views

How does he get this equality? [closed]

Let $u \in C^\infty_0(\Omega)$, fix $i = 1$. We have $$u(x_1, \ldots, x_n) = \int_{-\infty}^{x_1} \frac{\partial u}{\partial t}(t, x_2, \ldots, x_n)\, dt. $$ How do i do this integration? didnt ...
2
votes
4answers
84 views

multivariable limit of $\frac{x^2-y^2}{\sqrt{x^2+y^2}}$

Calculate multivariable limit of $$\lim_{(x,y) \rightarrow (0,0)}\frac{x^2-y^2}{\sqrt{x^2+y^2}}$$ How to do that? I was trying to figure out any transformations e.g. multiplying by denominator but I ...
1
vote
1answer
15 views

How to prove $\oint_{\partial D} \left( u \frac{d u}{d x} dx - u \frac{d u}{d y} dy \right) = \oint_{\partial D}u\frac{d u}{d \mathbf{n}} ds$?

For some simply connected region $D \subset \mathbb{R}^2$ with boundary $\partial D$, length differential along boundary $ds$ and normal $\mathbf{n}$, and some sufficiently smooth function $u(x,y)$ ...
-1
votes
0answers
23 views

Differentiability of CDF

I have joint pdf function $f_x(x)$ and $f_y(y)$ where $x \in [-a,a]$ and $y \in [-b, b]$ they are independent and identically distributed. I want to know that if PDF exists such that $$ ...
2
votes
1answer
33 views

Is it possible to have $|f(x) - f(y)| \leq M\| x - y \|$ under such conditions?

Let $f: A \to \mathbb{R}$ be differentiable on an open convex $A \subset \mathbb{R}^{n}.$ If $\| \nabla f \| \leq M$ on $A$ for some $M > 0,$ is it possible to have $$|f(x) - f(y)| \leq M \| x - y ...
0
votes
1answer
6 views

Multivariable Integral (changing coordinates)

I am trying to evaluate the following integral: $\int_{0}^{6}\int_{-\sqrt{36 - x^2}}^{\sqrt{36 - x^2}}\int_{-\sqrt{36 - x^2 - z^2}}^{\sqrt{36 - x^2 - z^2}} \dfrac{1}{\sqrt{x^2 + y^2 + z^2}} \ dy dz ...
0
votes
0answers
19 views

Given a function $x^a y^{1-a}$, confusion about how the curves vary as I very $\alpha$.

Suppose I have the function $$F(x,y) = x^a y^{1-a} = 1$$ where $a \in [0,1]$. $$g(x,y) = - \frac{\partial F /\partial x}{\partial F / \partial y} = \frac{a}{1-a} \frac{y}{x} $$ $$\frac{\partial ...
0
votes
0answers
14 views

Second partial derivative of a minimum function

I am reading a book on detection and estimation theory, and the author had this to say in the derivation of the white noise process from the Wiener process: We can formally obtain the covariance ...
2
votes
0answers
25 views

Tangent line of a lemniscate at (0,0)

I need to find the tangent line of the function $y=g(x)$ implicitly defined by $(x^2+y^2)^2-2a^2(x^2-y^2)=0$ at $(0,0)$, but I don't know how. I can't use implicit differentiation and evaluate at ...
0
votes
3answers
38 views

If $\nabla f(x) = 0$ on a set $S$, is $f$ necessarily constant on $S$?

If $\nabla f(x) = 0$ on a set $S$, is $f$ necessarily constant on $S$? I know it is true when $S$ is open convex, or open connected, but what about any arbitrary $S$?
3
votes
3answers
88 views

Why do lagrange multipliers have the form $\nabla G$

I was studying some multivariable Calculus and we were covering the topic of Lagrange multipliers. I didn't understand exactly why the equations take the form: $$ \nabla f = \lambda \nabla G $$ ...
0
votes
1answer
25 views

Basic Multivariable Question

I am working on a WebWork question which asks for the following: Find the volume between the cone $y = \sqrt{(x^2 + z^2)}$ and the sphere $x^2 + y^2 + z^2 = 4.$ I think I have worked out the ...
0
votes
1answer
48 views

What am I doing wrong here? Calculate the line integral of $f(x,y,z) = xe^{yz}$ from $(4,2,3)$ to $(0,0,0)$.

On the one hand, I should be able to calculate the integral from $(0,0,0)$ to $(4,2,3)$ and multiply that answer by negative one. Parameterize the line as $r(t) = (4t,2t,3t)$ for $0 \leq t \leq 1$. ...
0
votes
1answer
16 views

Chain rule, directional derivative - multivariable calculus

I am having a difficult time to understand the chain rule, and I have this exercise: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ to be a differentiable function. Define $\psi(x,y)=f(xy,x^2y^2)$. How ...
0
votes
0answers
10 views

ANSML - Proving of the matrix identity $\nabla_AtrABA^TC = CAB+C^TAB^T$

(ANSML is a tag I would like to use for Andrew Ng's Stanford Machine Learning - 2008) In this course, there were four matrix identities that I would like to prove. \begin{align} \nabla_a \text{tr}AB ...
1
vote
3answers
47 views

Does the limit exist? (Calculus)

Consider the function $$f(x,y)=\frac{2xy^2\sin^2(y)}{(x^2+y^2)^2}.$$ Does the limit exist when $(x,y)$ tends to $(0,0)$?
5
votes
1answer
56 views

Compute $\iiint_V \sin^2 (x + y + z) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}x$ where $V$ is an ellipsoid.

By performing a suitable scaling and rotation of the coordinates, or otherwise, evaluate $\iiint_V \sin^2{(x + y + z)}\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}x$ where $V$ is the region ...
1
vote
1answer
13 views

Remainder Taylor series two+ variables

Hard to find an article / tutorial specifically about this subject online.... Can anyone explain / send link to explanation or tutorial regarding how to calculate remainder for multi-variable taylor ...
0
votes
0answers
7 views

Boundary curve C orientation when surface is oriented in the direction of the positive y-axis.

I am currently working through some Stoke's Theorem problems. One problem in particular, as described below, has been troubling me because the solution manual states that the parametrization of the ...
0
votes
0answers
18 views

Eliminating parameters to find cartesian equation?

I have vector equation $r(u.v)=(x+a_1u+b_1v)i+(y+a_2u+b_2v)j+(z+a_3u+b_3v)k $ How can I eliminate u and v to get cartesian equation?
1
vote
1answer
39 views

What are some good books on vector analysis in higher dimenesion

What is some good books specifically on vector analysis in higher dimension? Standard vector calculus book usually only introduced double and triple integral method
0
votes
1answer
17 views

probability of joint PDF

I found $k = 4$ and yes, the are independent. But for the last one I know how to find the probability if they are like $x$ from $0$ to a number and $y$ from $0$ to a number so the limit of double ...
3
votes
0answers
39 views

Working out the area of Australia through Calculus? [closed]

I was wondering if it would be possible, and if so how, to calculate the area of an abstract shape on a sphere using surface integrals and Parametric surfaces and such. I am looking in to this as ...
1
vote
2answers
71 views

Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?

Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$? I've been attempting this with Lagrange multipliers in a few different ways. However, the ...
2
votes
1answer
26 views

Limit of a MultiVariable Equation

Hey guys I have a problem that asked to: Evaluate the following limit: $$\lim_{(x, y) \to (0, 0)} f(x,y) = \frac{2x^2y}{x^4+y^2}$$ along $y = mx$ and $y = mx^k$ I understand how to calculate the ...
4
votes
1answer
46 views

Lagrange's multiplier not working

Given the function $f(x,y):=xy+x-y$. Let $D:=\{(x,y)\in\mathbb{R}^2:x^2+y^2\leq25\wedge x \geq 0\}$. Find the absolute maximum and minimum of $f$ on $D$. My working is as follows: $\begin{array} ...
0
votes
2answers
92 views

online software to draw generic graphics for calculus and analysis?

could you mention any software in which I could design images like this.
0
votes
1answer
63 views

What is the shortest/longest distance from $9x^2 + 4y^2 = 36$ to $(5,5)$? [duplicate]

What is the shortest/longest distance from $9x^2 + 4y^2 = 36$ to $(5,5)$? Using Langrange Multipliers, I've set up the standard equation with $$g(x,y) = (x/2)^2 + (y/3)^2 = 1$$ $$f(x,y) = ...
1
vote
1answer
23 views

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$. If the directional derivatives are continuous, does this mean $f$ is differentiable?

There is a result which states that for a function $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$ if all its partial derivatives exist and are continuous then the total derivative $Df$ exists. If I ...
2
votes
2answers
52 views

How do I prove that $\alpha/\lVert\alpha\rVert$ is differentiable?

Let $\alpha\colon I\rightarrow \mathbb{R}^3$ be a $C^2$-curve such that $\alpha(t)\neq 0$. How do I prove that $\alpha/\lVert\alpha\lVert$ is differentiable?
3
votes
0answers
40 views

Differential Form Pullback Definition

I'm having some difficulty following how Spivak (Calculus on Manifolds) has induced the pullback on page 89. From reading elsewhere online it seems convention is to define the induced map of the ...
0
votes
0answers
20 views

Question about the chain rule of matrix calculus?

From the wiki page matrix calculus, I learned the chain rule for matrix calculus is that (assuming numerator layout) ...
0
votes
4answers
32 views

Distance to origin from curve

Hello all I am trying to redo a problem I had and I am stumped for some reasons. I just want to find the maximum and minimal distance from the curve $$7x^2-6xy+7y^2-6=0$$ to the origin. But I want to ...
1
vote
2answers
28 views

Straight vs Partial derivative

Does it make sense to write $\frac{d}{dx}u(x,t)$ or can one only write $\frac{\partial}{\partial x}u(x,t)$?
-2
votes
1answer
22 views

Prove $M$ is a manifold [closed]

Let $ M$ be the set of all points $(x, y, z) \in \mathbb{R}^3$ satisfying both of the equations $x^2 + y^2 = z^2 + 1$ and $x + y + z = 0$. Prove that $M$ is a manifold. What is the dimension of $M$?
1
vote
1answer
36 views

Seeking Recommendation on Theoretical Multivariable Calculus textbooks

I am a college sophomore with double majors in mathematics and microbiology. I wrote this email to seek your advice on selecting a theoretical, proof-based textbook on the multivariable calculus. I ...
0
votes
0answers
19 views

multivariable linearization

I have been asked to linearise the fallowing equilibrium points are phy=theta yaw=0 x,y,z=0 The idea I have using V'z as a model: -g+(kcm/m)(cos(phy)cos(thata)*voltages + ...
0
votes
0answers
31 views

Multivariable Chain Rule: Finding ∂z/∂y and ∂z/∂x

Question: The equation $$7xyz=2x^2+y^2+3z^2+7$$ implicitly defines z as a function of x and y in the neighborhood of the point where $x=2, y=1$ and $z=2$ . Find ∂z/∂x and ∂z/∂y at this point. ...
0
votes
0answers
13 views

Variant of Stokes theorem

I have recently, for the first time, seen the following version of Stokes' theorem stated: For a region $R$, $d\nu$ a volume element, $ds$ a surface element and $n(x)$ the unit normal to the surface ...
0
votes
0answers
15 views

Numerical Triple integral with three other parameters in R

I am trying to integrate this function $f(u,v,w; t,x_{0},z)$ with respect to three variables, $u$, $v$, $w$, although the function also have other three parameters $t$, $x_0$, and $z$. Question: How ...
-2
votes
0answers
22 views

Consider the function $f(x, y) = \arctan(x + 3y)$ [closed]

Consider the function $f(x, y) = \arctan(x + 3y)$. (a) Write down the equation for the linear approximation L(x, y) to f(x, y) at the point (x, y) = (1, 0). (b) Derive the quadratic approximation ...
0
votes
2answers
34 views

Integrating the derivative of a multivariate function

If $u=u(x,t)$ how do I compute $\displaystyle \int \frac{du}{dt}dt$ ? Would I be correct in saying it is not simply a case of cancelling the $dt$'s and getting an answer of $u$+ constant, seeing as ...
1
vote
2answers
41 views

Applying Chain rule to $z = z(u, v) = f(x(u, v), y(u, v))$.

If $z = z(u, v) = f(x(u, v), y(u, v))$ is a differentiable function, where $x = x(u, v) = u^2 − v^2$, $y = y(u, v) = 2uv$, show that $$\frac{∂^2f}{∂x^2} +\frac{∂^2f}{∂y^2} =\frac{1}{4(u^2 + ...
0
votes
2answers
58 views

Prove integrable function by partition

Let $f(x):[0,1] \rightarrow \mathbb R$, $$f(x)=\begin{cases} 1 & \text{ if } x=1/n ,\text{ $n$ is an integer }\\ 0 & \text{ otherwise }\end{cases}$$ a) Prove that $f$ is integrable b) Show ...
0
votes
0answers
10 views

Does the total derivative always require its inputs (e.g. $x,y$) to be defined parametrically (e.g. $x(r,\theta)$)?

Suppose we have $f(\theta,x(r,\theta),y(r,\theta))$ with appropriate conditions on $f$ such that we can take a total derivative. Then, the total derivative with respect to $\theta$ is $$\frac{\rm{d} ...
0
votes
3answers
16 views

Linear Approximation: Find the linear approximation at a point

Question: f(x,y)=√(7+2xy) Find the linear approximation at (3, -1) My answer: So I took the partial derivative at x and y, and I got Fx = y/(√(7+2xy)) and Fy = x/(√(7+2xy)). Evaluating for this, I ...
2
votes
1answer
23 views

Multi Variable Limit

Can anyone show me the steps? The limit is $0$ but I am facing some difficulties in getting to that point! I know that $\ln(1+u) \leq u$ for $u>-1$. $$\lim_{x,y\to ...