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
0answers
15 views

When can the Second Derivative test for functions of 2 variables not be used?

For this test $d = f_{xx}(a,b)f_{yy}(a,b)-f_{xy}(a,b)^2$ if $d>0$ then $(a,b)$ is a max or min. This can be determined if $f_{xx}$ or $f_{yy}$ is negative or positive because they will both be ...
0
votes
1answer
31 views

Verifying stationary points - check my answer please - Has a hessian

Can someone check this for me; For $f(x_1,x_2,x_3) = x_1^2 + x_2^2 + x_3^2-x_1x_2+x_2x_3-x_1x_3-x_1+x_2$ the stationary point occurs at $\nabla f(x)^T =\left[ \begin {array}{c} 0\\ 0\\ 0\\ ...
0
votes
1answer
38 views

Why should the derivative of a parametrization to a manifold be one to one?

Hubbard and Hubbard define the parameterization of a manifold $M \subset \Re^n$ as a mapping $\gamma : U \subset \Re^k \to M$ satisfying $U$ is open $\gamma$ is $C^1$, one to one and onto $M$ ...
0
votes
1answer
49 views

Converting an integral from polar to Cartesian. Trouble finding limits.

I am trying to convert the integral below into Cartesian form. However, I'm having trouble finding what the limits should be. The integral should evaluate to $\frac12\pi(b^4-a^4)$. For background, ...
1
vote
1answer
127 views

Derivative of trace of inverse matrix?

I've been trying to derive the formula for the derivative of $Tr(X^{-1})$ w.r.t. $X$, which I know is $X^{-2T}$. According to the Matrix Cookbook $$\dfrac{\partial g(U)}{\partial X_{ij}} = ...
0
votes
1answer
33 views

Evaluate all first order partial derivative of $f(x,y,z)=x^\frac{y}{z}$

Evaluate all first order partial derivative of $f(x,y,z)=x^\frac{y}{z}$ Here is my attempt: $y$ and $z$ are constant, so we use the power rule to evaluate $f_x$: $$ f_x = ...
2
votes
1answer
34 views

Use implicit differentiation to determine partial derivative

Use implicit differentiation to determine $\frac{\partial z}{\partial x}$ in $yz=ln(x+z)$ and $ \sin(xyz)=x+2y+3z$. Here is my answer: $$ yz=ln(x+z) $$ $$ yz'=(1+z')\frac{1}{x+z} $$ $$ z' = ...
1
vote
1answer
34 views

Determine a parameterization for the line which is tangent to the curve at t=2

(1) A curve is given by the function $$r(t)=(t^3 -3t^2 +2t +4)i + (13-5t)j +(t^2 -t-3)k$$ Determine a parameterization for the line which is tangent to the curve at $t=2$ I started by solving for ...
0
votes
0answers
22 views

Piecewise continuous potential function?

This question arises from a game theoretic problem. I'm posing it in 2-D to simplify the exposition. Given two functions $f(x,y)$ and $g(x,y)$, the problem is to find a potential function $P(x,y)$ ...
0
votes
1answer
20 views

Multi-variable particle question

A particle is initially at rest at the origin $(0,0,0)$. At time $t=0$ it begins moving with velocity: $$V(t)= (cos(t)-tsin(t),1,sin(t)+cos(t))$$ find the time $t$ at which the particle leaves the ...
1
vote
1answer
574 views

Does integration by parts work for partial derivatives?

Does integration by parts works for partial derivatives? Can we write $$\int_a^b \frac{\partial f(x,y)}{\partial x}g(x,y) dx = f(x,y)g(x,y)|_a^b - \int f(x,y)\frac{\partial g(x,y)}{\partial x}dx$$
1
vote
1answer
47 views

Calculating joint cumulative distribution function

given the joint probability density function $f_{XY}(x;y)= \begin{cases} 1, & \text{if (x; y) $\in[0; 1]\times[0; 1]$} \\ 0, & \text{elsewhere} \end{cases}$ I want to calculate the joint ...
0
votes
0answers
23 views

Setting up integral using Spherical Coordinates

A sphere of radius $2$ is cut $1$ unit from the center of the sphere, leaving one small part and one large part. Set up an integral using spherical coordinates to find the volume of the smaller part. ...
1
vote
1answer
39 views

An exercise about immersion and curves

Let $\alpha : \mathbb{R}^2 \to \mathbb{R}^3$ a immersion in class $C^1$, injective and had his inverse $\alpha^{-1} : \alpha(\mathbb{R}^2) \to \mathbb{R}^2$ continuous. Let $\gamma : \mathbb{R} \to ...
1
vote
1answer
28 views

Proving an inequality of two variables

The inequality is the following : $\frac{x^2+y^2}{4} \leq e^{x+y-2}$, where $x,y \geq0$. I tried manipulating the inequality using Taylor series, but I couldn't find a conclusive result . Any ideas ...
0
votes
0answers
11 views

Differentating Series of Functions

I have a function: $$x: \mathbb{Z}\times\mathbb{R}\to\mathbb{R}$$ $$x(t,w) = \begin{cases} f(w\: x (t-1,w)) & if\: t>0 \\ 0 & if\:t<=0 \end{cases}$$ I would like to find ...
0
votes
1answer
85 views

Finding a function ,satisfying the given properties

Finding a function of two variables , satisfying $$\lim_{\left(x,y \right)\rightarrow \left({x}_{0},{y}_{0} \right)}f\left(x,y \right)=+\infty ;$$ and for $\forall\delta>0, \exists y^{'},y^{''}\in ...
5
votes
5answers
122 views

Does $\left(\frac{\partial f}{\partial x}\right)^2=\frac{\partial^2f}{\partial x^2}$

This may be an obvious question but I'm just not thinking straight, thanks The answer must be no
3
votes
2answers
22 views

What vectors in the plane z=x are orthogonal to $(1,-1,0)$?

I believe all such vectors should be of the form $(a,b,a)$. Hence, $$ (a,b,a) \cdot (1,-1,0) = a - b + 0 = 0 \implies a=b$$ So all the vectors we seek are of the form $$ (a,a,a)$$ where $a \in ...
0
votes
1answer
19 views

Which of the surfaces does the vector lie on?

So I used the trig identity (y^2 + z^2 = 1) on my y and z component. So I concluded that the cylinder y^2 + z^2 = 4 satisfies the question. I also concluded that the plane x + y = 3 satisfies the ...
1
vote
1answer
21 views

Multi-variable calculus involving $\ln$

I am having difficulty with differentiating this equation with respect to $y$: $$ W= x^{y \ln(z)}. $$ Differentiating calculators are giving me the answer $$\ln(x) \ln(z).x^{y \ln(z)}$$ But I ...
0
votes
1answer
17 views

If $\xi=kx$, does $\frac{d^{2}}{dx^{2}}=\frac{d^{2}}{d\xi^{2}}$

Let $\xi= kx$. Does $\dfrac{d^{2}}{dx^{2}}=\dfrac{d^{2}}{d\xi^{2}}$? If so, why? If not, what factor do I need to account for to make this change of variables?
1
vote
2answers
25 views

What vectors are orthogonal to $(3,2,-5)$ and have components that sum to 4?

I have so far that $(3,2,-5) \cdot (x,y,z)= 3x + 2y -5z =0$ and $x+y+z=4$ must be true. Combining these equations gives us $4x+3y-4z=4$. If this is the right track, I'm not sure where do I take it ...
0
votes
0answers
56 views

Coordinates at a singularity

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a smooth function. Let's assume that $f$ has a local minimum at $p \in \mathbb{R}^2$ and hence $| \nabla f|\ = 0$ at $p.$ Intuitively, one should be able to ...
1
vote
1answer
25 views

How do I correctly introduce a time parameter into this equation?

So, for the past few years it's been my goal to create an equation that would give me the position of an object in a gravitational field at time $t$, given it's initial position and velocity. At first ...
0
votes
0answers
16 views

move forward with quadratic rates of change relationship

I am examining the relationship between two time-sensitive variables, $f_1(t)$ and $f_2(t)$. If I plot $df_1/dt$ agains $df_2/dt$, I find a parabolic line, $df_2/dt = k - (df_1/dt)^2$ where k is a ...
1
vote
2answers
775 views

Find the mass of the disk. - Double Integration Problem - Calculus 3

A disk of radius 5 cm has density 10 g/cm2 at its center, density 0 at its edge, and its density is a linear function of the distance from the center. Find the mass of the disk. my answer: 157.08g ...
0
votes
0answers
39 views

Divergence theorem: 2 different results

Given the vector field $\vec a(r, \theta, \phi) = \frac3r \vec e_r = \frac3r \begin{pmatrix} \cos(\phi) \sin(\theta) \\ \sin(\phi) \sin(\theta) \\ \cos(\theta)\end{pmatrix}$, I have to calculcate the ...
0
votes
2answers
52 views

The total derivative

I am having a hard time trying to figure out the meaning of the total derivative. So, please help me in understanding its geometrical aspects. Like how do I see geometrically whether the total ...
0
votes
0answers
19 views

Pre-requisites for learning multivariable calculus

I've been interested in learning about multivariable calculus now. What topics do you suggest that I familiarise myself with before I dive into it? Also if possible, can you provide any recommended ...
1
vote
1answer
29 views

Prove that the Inverse Parametric equation is continuous.

Original question: Show that a cylinder {$(x,y,z) \in R^3 : x^2+y^2=1$} is a regular surface, and find parametrization whose coordinate neighborhoods cover it. First of all, what do it mean to ...
-4
votes
3answers
97 views

How Can I find partial derivative of integral function with respect to x?

I am given $$f(x,y) = \int_{0}^{\sqrt{xy}} e^{-t^{2}} dt$$ For $x, y > 0$ How can I find partial derivative of $f$ with respect to $x$ ? I am trying to integrate it first but it is something ...
0
votes
0answers
29 views

Find the extension of Euler's theorem for homogeneous functions of degree p in 2 dimensional case

What is the extension of Euler's theorem for homogeneous functions of degree p in 2 dimensional case? and How can I prove this? It seems little bit awkward , since for 2 dimension ...
0
votes
0answers
68 views

Euler's theorem of homogeneous function (I dont understand the proof)

Here is a link for those who want to take a look at the theorem. (http://people.hss.caltech.edu/~kcb/Notes/EulerHomogeneity.pdf) I considered a function g(t)=f(tx) for fixed x and I took a ...
0
votes
0answers
48 views

Proof technique question related to Euler's theorem for homogeneous function

I am trying to prove Euler's theorem for homogeneous function. Actually, proof is given http://people.hss.caltech.edu/~kcb/Notes/EulerHomogeneity.pdf What I don't understand is that proof is based ...
1
vote
1answer
30 views

Solving for one of variables locally when the Implicit Function Theorem does not apply

I'm having trouble deciding whether certain functions can be locally solved. I have some examples: Can $xye^{xz} - z\log y =0$ be locally solved in $(0,1,0)$ for x? y? z? In this case, I used ...
1
vote
1answer
136 views

Volume of $y = 6\sqrt{\sin(x)}$ rotated around $y$-axis using triple integrals

The problem is to find the volume of $y = 6\cdot \sqrt{\sin (x)}$ rotated around the $y$-axis when $0 \leq y \leq 6$. I know this can be done by the sv-calc method of volumes of revolution but I ...
1
vote
1answer
24 views

Finding the line of intersection of two planes

This is a problem on my homework (so please do not provide more than hints as I definitely don't want you to do it for me). I'm just stuck and was hoping someone might point out my mistake or suggest ...
2
votes
2answers
184 views

Limit of a function with two variables: when do we stop looking for another value?

So for instance we have this limit of a function $\displaystyle\lim_{(x,y)\to(0,0)}{xy\over \sqrt{2x^2+y^2}}$, and the function isn't continuous at the point $(0,0)$. Now we can try to find the limit, ...
2
votes
0answers
46 views

How do you calculate the directional derivative in multiple dimensions?

I've seen lots of examples of calculating the directional derivative where the solution takes the total derivative and then calculates the dot product with the unit directional vector. However this ...
-1
votes
0answers
14 views

Plotting functions of 3 variables

Functions of two variable, we can plot in 3D, making the height the result of the function, or we can project them onto a lower, second, dimension in the form of level curves. But we do not really ...
0
votes
1answer
106 views

Multivariable Calculus - Object movement and rate of change of its temperature problem

An insect is moving on the ellipse $2x^2 + y^2 = 3$ on the xy-plane in the clockwise direction at a constant speed of 3 centimeter per second. The temperature function $T(x; y)$ (experienced by the ...
2
votes
0answers
27 views

A converse theorem to $\operatorname{div} (\operatorname{curl}(\mathbf{F}))=0$

I have already proved the following: Let $\mathbf{F}$ be a vector field of class $C^2$ defined on $ \mathbb{R}^3$. If $\nabla\cdot\mathbf{F}=0$, then there is some vector field $\mathbf{G}$ such that ...
1
vote
2answers
168 views

Partial Derivatives and the Fundamental Theorem of Calculus

I am being asked to evaluate the 1st-order partial derivatives $-$ $f_{x}$(x,y) and $f_{y}(x,y)$ $-$ of the following multi-variable function: $f(x,y) = \int_{y}^{x} cos(-1t^2 + 3t -1) dt$. Any help ...
1
vote
0answers
39 views

How to apply the chain rule for partial derivatives to transformations?

I'm currently working to solve the Black-Scholes model partial differential equation (it's a model for a.o. stock option prices). The Black-Scholes equation for a calloption C(S,t) is given by $ ...
0
votes
2answers
30 views

Range of a parameterized surface function

Here is a question from my textbook and I have to find out its domain and range. $$\vec f(x,y)=(x+y , \frac{1}{y-1} , x^2+y^2) $$ I can get the correct domain but the answer of the range given by ...
0
votes
1answer
26 views

When are there infinite stationary points?

Say we are given: $$f(x_1,x_2) = \alpha (x_1^2+x_2^2) + \beta x_1x_2 + x_1 + x_2$$ then $$\nabla f = \langle f_{x_1}(x_1,x_2), f_{x_2}(x_1,x_2) \rangle = 0$$ for stationary points. which will give ...
0
votes
2answers
88 views

Limit as epsilon approaches 0

Could anyone shed some lights on this problem: what's the limit of $y(x,a,\varepsilon)$ when $\varepsilon \to 0$ $$\def\pow#1{e^{#1/\varepsilon^2}} y(x,a,\varepsilon) = ...
0
votes
0answers
46 views

Evaluating the divergence theorem for region above $z=0$, below $z=x$ and inside $x^2+y^2=1$ where $\hat F=(xz,yz,z^2)$

Can someone please confirm my working below: The answer am getting look kinda crazy -Thanks. $$\color{green}{\hat F=(xz,yz,z^2)}$$ $1.$For the surface where $\color{green}{z=0}$ i.e. (flat ...
2
votes
2answers
88 views

Can you have a gradient of time?

Okay this maybe a very stupid question but in my calculus III class we introduced the gradient but I am curious why don't we also include the derivative of time in the gradient. Thanks, math noob