Multivariable calculus help?!?
Use Lagrange Multipliers (calling the constraint g). Then, ∇f = λ∇g ==> <(1/n)(x₁x₂ ... xn)^(1/n - 1) * (x₂ ... xn), ...,(1/n)(x₁x₂ ... xn)^(1/n - 1) * x₁x₂ ... x(n-1)> = λ<1, 1, ..., 1>. Equating like entries: (1/n)(x₁x₂ ... xn)^(1/n - 1) * (x₂ ... xn) = λ ... (1/n)(x₁x₂ ... xn)^(1/n - 1) * (x₁x₂ ... x(n-1)) = λ Rewrite these as follows: (1/n)(x₁x₂ ... xn)^(1/n - 1) * (x₁x₂ ... xn)/x₁ = λ ... (1/n)(x₁x₂ ... xn)^(1/n - 1) * (x₁x₂ ... xn)/xn = λ So, we have (1/n)(x₁x₂ ... xn)^(1/n) / x₁ = λ ... (1/n)(x₁x₂ ... xn)^(1/n) / xn = λ Hence, λ / [(1/n)(x₁x₂ ... xn)^(1/n)] = x₁ = ... = xn. -------- Since x₁ = ... = xn, substituting this into g yields nx₁ = c ==> x₁ = c/n. Hence, (x₁, ..., xn) = (c/n, ..., c/n) is the critical point, and f(c/n, ..., c/n) = c/n is the maximum value, since (c, 0, ..., 0) satisfies g, but f(c, 0, ..., 0) = 0 < c/n. --------- Therefore, f(x₁, ..., xn) ≤ f(c/n, ..., c/n) for all (x₁, ..., xn) with x1, ..., xn > 0 ==> (x₁ ... xn)^(1/n) ≤ c/n for all (x₁, ..., xn) with x1, ..., xn > 0 ==> (x₁ ... xn)^(1/n) ≤ (x₁ + ... + xn)/n for all (x₁, ..., xn) with x1, ..., xn > 0, via definition of g. I hope this helps!
Multivariable Calculus Help :/?
Hi Alfredo, The first thing to do is to consider the bounded region A, to try and gauge what the function might look like, and if there are any singularities in the range. Fortunately, begin in the first quadrant, and bounded by the x- and y-axes, it is easy to picture the function. Moreover, f(x,y) = 5xy is a simple function, with no singularities to worry about. So, all we need to do is to define the region A in the first quadrant. This is done by considering the equation y = √(9 - x²). On squaring, we have y² = 9 - x², or x² + y² = 9. This is a circle equation, of centre (0,0) and radius 3. So, this is how our function is bounded. This is important because it gives us the x-limit for integration, x = 3. To find the volume, simply set up the double integral, using the y-limits as the axis, and the curve boundary function, and the x-limits as the axis, and 3: V = ∫ {x = 0 to 3} ∫ {y = 0 to √(9 - x²)} 5xy dy dx. Do the inner integral in y first, treating x as a constant: V = ∫ {x = 0 to 3} [5xy² / 2] {y = 0 to √(9 - x²)} dx. Apply the limits: V = ∫ {x = 0 to 3} [5x(9 - x²) / 2] dx. Simplify the integral: V = (5 / 2) ∫ {x = 0 to 3} 9x - x³ dx. Integrate with respect to x: V = (5 / 2) [9x²/2 - x⁴/4] {x = 0 to 3}. Apply limits: V = (5 / 2) [9*3²/2 - 3⁴/4]. Calculate powers: V = (5 / 2) [81/2 - 81/4]. Simplify square bracket: V = (5 / 2) [81/4]. Multiply together: V = 405 / 8. Add units in three dimensions: V = (405 / 8) (units)³. ----- Solution: ----- {2} V = (405 / 8) (units)³.
What is the best book for learning multivariable calculus?
Considering that many of the most important results of multivariable calculus were originally discovered and developed in the context of electricity and magnetism, the best book for learning multivariable calculus is also the best book for learning electricity and magnetism: Electricity and Magnetism (Berkeley Physics Course, Vol. 2): Edward M. Purcell: 9780070049086: Amazon.com: BooksI am being totally serious. This is the best way that you can understand what Stokes's theorem and Gauss's theorem are really about, at least at the undergraduate level.
Help with calculus please multivariable?
∫c (2xy dx + 0 dy) = ∫∫ (∂/∂x)0 - (∂/∂x)(2xy)] dA, by Green's Theorem = ∫(x = 0 to 1) ∫(y = 0 to 6) -2y dy dx = ∫(x = 0 to 1) -y^2 {for y = 0 to 6} dx = -36. I hope this helps!
Question about multivariable calculus. please help?
A cone C with height 1 and radius 1, has its base on the xz-plane and its vertex on the positive y axis. Find a function g(x,y,z) such that C is a part of the level surface g(x,y,z) = 0. [Hint: The graph of f(x,y)=sqrt(x^2+y^2) is a cone which opens up and has vertex at the origin.]
How much help does taking multivariable calculus in high school when applying to elite schools like MIT, Stanford etc.?
It helps more for MIT than Stanford. It helps for MIT especially if taking multivariable calculus is uncommon for students at your high school because it shows self-initiative and mathematical interest.Out of all the top schools, MIT cares the most about highschool courseware. On the MIT application, it actually asks for you to list all the courses in each subject that you’ve taken. The MIT application also asks in what year you took Calculus, Physics, Chemistry, and Biology (this is to make sure you are prepared for the General Institure Requirements (GIRs) that all MIT students are required to pass).MIT is pretty rigorous and majoring in STEM can be very challenging. This means that MIT vets students for academic preparation more intensely to make sure that students will be able to handle their coursework. Taking Multivariable and getting an A in the class is a great way to show that you will be able to handle MIT - I would definitely recommend it.In addition, I would recommend that you take Discrete Math, Linear Algebra, and Differential Equations if you can. If you get into MIT, you’ll thank me later :)That being said, if you are interested in applying to a top school and getting personal advice from me, a friend and I are running a site where we give feedback for students at Applihood.com.
What should you know before starting multivariable calculus?
The most important and pretty much only prerequisite for learning multivariable calculus is having a solid grasp on single variable calculus.Also, visualizing in 3 dimensions is quite important in several topics of multivariable calculus. This is a little tougher than most people think as representing 3 dimensions on a piece of paper or a computer screen requires converting it to a 2-dimensional figure.Some basic vector algebra might be useful (even if you're not interested in vector calculus).I can't say much about linear algebra as my knowledge in that is quite limited. However, determinants are used in multiple places throughout multivariable calculus so you should at least know what a determinant and a matrix are and some basic properties.Keep in mind that multivariable calculus is all about taking a problem and converting it to an equivalent single variable calculus problem (roughly speaking). There's not nearly as many new concepts as in one-variable calculus and it is mostly about generalizing properties of functions that take one argument to functions that take multiple arguments.
Help with university level multivariable calculus please?
Suppose that the equation F(x,y,z)=0 implicitly defines each of the three variables x,y, and z as functions of the other two: z=f(x,y), y=g(x,z), x=h(y,z). If F is differentiable and Fx, Fy, and Fz (the partial deviations) are all nonzero, show that (dz/dx) . (dx/dy) . (dy/dz) = -1 Any help will be greatly appreciated!! Please and thank you! However, I must say at the outset that this question is at university/college level, so unless you're really smart or are ahead of your time, I gently advise you not to answer if you're under 19 years old. Thank you!
What are the best resources for mastering multivariable calculus?
See Multivariable calculus learning recommendations