Multivariate calculus Definition and 31 Threads

Hello. I am currently doing a high school univariate calculus book, but I would like to go through Apostol's two volumes to get a strong foundation in calculus. His first volume seems great, and I've heard great things about his series, but I am not sure if his second volume contains sufficient...

I just finished multivariate calculus (without any linear algebra experience yet) and I am seeking out a path to understanding General Relativity. I am wondering what are the mathematical fields after multivariate calculus that I need to master before beginning to understand GR, and what...

I do not understand how can I parameterize the surface and area and line differentials.

1. We find the partial derivatives of ##f## with respect to ##x## and ##y## to get ##f_x = \frac{2\ln{(x)}}{x}## and ##f_y = \frac{2\ln{(y)}}{y}.## This makes the gradient vector $$\nabla{f} = \begin{bmatrix} f_x \\ f_y \end{bmatrix} = \begin{bmatrix} \frac{2\ln{(x)}}{x} \\ \frac{2\ln{(y)}}{y}...

Afternoon, I have been working on this problem for awhile now but have been stuck on a certain point, and once I set everything equal to each other I end up with the same thing for example: x=x, y=y, & z=z Find the minimum and maximum values of the function f(x,y,z)=3x+2y+4z subject to the...

I have been stumped on this problem and I am probably overthinking it as usual. The problem deals with the Lagrange Multipliers: Use Lagrange multipliers to find the point (a,b) on the graph of y=e^(9x), where the value (a,b) is as small as possible. I have found the gradient for both when...

Okay, I never took calculus 3 so I don't know how to write/solve an integral for a power I made up. I want to flesh out my game world with math because it's cool and I can. Like Chester wrote some of the physics for the Bizarro universe. Applying physics and math to a fictitious universe is...

Homework Statement Griffiths Introduction to Electrodynamics 4th Edition Example 1.10 Check the divergence theorem using the function: v = y^2 (i) + (2xy + z^2) (j) + (2yz) (k) and a unit cube at the origin. Homework Equations (closed)∫v⋅da = ∫∇⋅vdV The flux of vector v at the boundary of the...

I've taken the single and multivariate calculus classes at my school, (college class offered in my high school for accelerated students). I'm currently a junior, but in the summer of my senior year, I plan to read a book on proofs, brush up on math glossed over by the american education system...

Let us assume the validity of Ampère's circuital law\oint_{\gamma}\mathbf{B}\cdot d\mathbf{x}=\mu_0 I_{\text{linked}}where ##\mathbf{B}## is the magnetic field, ##\gamma## a closed path linking the current of intensity ##I_{\text{linked}}##. All the derivations of the Biot-Savart law for a...

Dear Physics Forum advisers, Could you recommend books that treat the multivariable calculus from a theoretical aspect (and applications too, if possible)? I have been reading Rudin's PMA and Apostol's Mathematical Analysis, but their treatment of vector calculus is very confusing and not...

I am trying to understand multivariate linear regression. I have a list of time that it took running processes based on several params, like % of cpu usage, and data read. Eg, I have a process that took 50 seconds to run, with a cpu usage of 70%, and the process read 10bytes of data. I have...

Let f: \mathbb R^2 \to \mathbb R^2 given by f=(sin(x-y),cos(x+y)) : find the equation of the tangent plane to the graph of the function in \mathbb R^4 at (\frac{\pi}{4}, \frac{\pi}{4}, 0 ,0 ) and then find a parametric representation of the equation of the tangent plane What I did: the...

Hello, I would like to differentiate the following expected value function with respect to parameter $$\beta$$: $$F(\xi_1,\xi_2) =\int_{(1-\beta)c_q}^{bK+(1-\beta)c_q}\int_{(1-\beta)c_q}^{bK+(1-\beta)c_q}\frac{\xi_1+\xi_2-2bK}{2(1-\beta)^2} g(\xi_1,\xi_2)d\xi_1 d\xi_2$$ $$g(\xi_1,\xi_2)$$ is...

Homework Statement Use technique of completing squares to Show that this function has an absolute minimum. f(x, y) = x^2 + y^2 − 2x + 4y + 1 Homework Equations Not entirely sure how completing the squares will indicate an absolute minimum.Is there some additional reasoning required? The...

1. The problem I am trying to prove the following relation in cartesian coordinates. We were given a hint to use integration by parts, as well as the fact that we know $d \vec r = dx\,dy\,dz$ (volume integral). $$\int f(\vec r)\ \nabla \cdot \vec A(\vec r) \, d \vec r = -\int \vec A(\vec...

Homework Statement I was given the following function f(x,y) = \begin{cases} \frac{x^2y}{x^4+y^2} & (x,y) \neq 0 \\ 0 & (x,y) = 0 \end{cases} Which of the following are true? (I) f is not continuous at (0, 0). (II) f is differentiable everywhere (III) f as a well defined partial...

Homework Statement Let f:R2−>R be a differentiable function at any point, and g be the function g:R3−>R2defined by: g(u,v,w)=(g1,g2)=(u2+v2+w2,u+v+w) consider the function h=fog and prove that ||∇h||^2 = 4(∂f/∂x)^2*g1 + 4(∂f/∂x)(∂f/∂y)*g2 + 3(∂f/∂y)^2.The Attempt at a Solution...

I have an optimization where I'd like to express the idea that some of my parameters cannot equal a certain value e.g... max \ f(x) = ... s.t. x_3 \neq 1 Is there a standard method to solve this using lagrangian optimization? Thanks.

Just started multivariate course, can't figure out this simple question. If f(u,v,w) is a function of 3 variables. And u, v and w are themselves function of t. Then does f(u,v,w)=0 implies df/dt=0 or df/du=0. or both.

A firm has the following total-cost and demand functions: C = aQ^3 - bQ^2 + cQ + d Q = e - P (d) Find optimizing level of Q. (e) Chooses a,b,c,d and e such that there is only one profit-maximizing level of output Q. I found 2 solutions for question d, but in a very long and messy...

It has been awhile since I've posted. :rolleyes: Anyways... I'd like to start learning multivariate calculus along with linear algebra. However, I have a little problem in choosing a suitable text. I'd like to have a text with more theory and less applications. Some people have recommend...

Homework Statement Consider the surface x^2 / 4 + y^2 + z^2 = 3 (a) what type of quadric surface is this? (a spheroid...) (b) find the outward unit normal vector to this surface at the point (2,1,1). Homework Equations How do we find a normal vector? Does one just take grad(f(2,1,1))...

Homework Statement Apply the two cases of th change rule. For example: The voltage V in an electrical circuit is slowly decreasing as a battery wears out and the resistance R is slowly increasing as the resistor heats up Use Ohm's law V=IR to find how the current is changing (with respect to...

I've spent a lot of time soul searching after having some academic "failures" in the previous semesters, and what I found is that often when I'm not studying, it's not because I can't study, but it's because the book we're using is unreadable to me. For example, after sailing through Calculus...

I wish to master Multivariate Calculus(to the extent required to get an A with a ridiculously difficult professor) and Linear Algebra in 3 weeks so I can breeze through both classes in the Fall semester. I've self studied both of my textbooks that I will be using for the two courses, but I'd...

Hi, I have this qualifying exam coming up in about a month for my Ph.D. program in math. I just noticed that my European undergraduate education covered nothing about actually calculating any tricky integrals. It was only about Riemann sums and proving properties of the integral which I know...

Homework Statement Show that: max{a,b}=1/2*(a+b+|a-b|) Hence or otherwise, show that if f,g:R ----> R are continuous, then the function h:R ----> R defined by h(x) = max{f(x),g(x)} is also continuos. Suppose the f:(0,inf) ---> R is differentiable and f(x) ---> 0 as x---> inf. Set...

Hi, could anyone tell me the steps to solve the following question: Find the solution of x'=Ax with the initial value -------1---------2 0 0 x(0)=( 0 ), if A=( 0 1 -1 ) -------1---------1 1 1

I am currently studying multivariate calculus, and quite frankly, I'm not doing well. I am currently using Thomas' Calculus, 11/e media upgrade by Pearson. I don't like this book. The text is 75% theory and 25% simple examples, too few exercise problems, and the explanation is just too difficult...

Homework Statement This is a second (university) year calculus problem dealing with calculus of multiple variables. In economics, utility is a measure of the relative satisfaction from, or desirability of, consumption of goods. A utility function u = u(a,b) gives the utility from consuming...