Multivariable Calculus Examples: Learn Complex Calculus Problems

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Calculus and Analysis
Multivariable calculus

Recent questions in Multivariable calculus

Gage Potter 2022-05-02

Solve for w in the equation y-1=4wy-6.

Jordin Olsen 2022-05-01

Solve for x in the equation 3-3x=w+1.

dificultehcm 2022-05-01

Solve for v in the equation -7=3uv+4.

broggesy9 2022-04-30

Solve for x in the equation a-7x+4=0.

Edith Mayer 2022-04-07

I want to ask whether the following question has a closed form solution:
${\begin{cases} min_{x} c^{T} x \\ s.t. \\ 1^{T} x = b \\ x \geq 0 \end{cases}$
We can write down the kkt condition as:
$\begin{aligned} L & = c^{T} x - ⟨ λ_{1}, 1^{T} x - b ⟩ - ⟨ λ_{2}, x ⟩ \\ \partial L / \partial x & = c - 1 λ_{1} - λ_{2} = 0 \\ [λ_{2}]_{i} \cdot x_{i} & = 0 \\ [λ_{2}]_{i} & \geq 0 \end{aligned}$
Denote ith component of $c$ as $c_{i}$ . Now we can see that when $(c_{i} - λ_{1}) \geq 0$ , then we must have $x_{i} = 0$ and $(c_{i} - λ_{1}) = 0$ , we also must have $x_{i} > 0$ . But I am not sure how to proceed to the next.

lasquiyas5loaa 2022-04-07

$U$ is a random variable that follows Uniform distribution with interval $(0, 1) . X_{1}, . . ., X_{n}$ follows Bernoulli distribution with mean $U$ . How do I find the function $f : (0, 1)^{n} - > R$ that minimizes $E [(U - f (X_{1}, . . ., X_{n}))^{2})]$ ? I have no idea where to start solving. Thank you in advance.

habib A2022-04-02

Equation of the line tangent to $\{\begin{cases} y = x^{2} \\ z^{2} = 16 - y \end{cases}$ , at the point (4,16,0) ?
a) $\{\begin{cases} 4 x = y \\ z = 0 \end{cases}$
b) $\{\begin{cases} x = 4 \\ y = 16 \end{cases}$
c) $\{\begin{cases} y - x = 12 \\ z = 0 \end{cases}$
d) $\{\begin{cases} x + y = 20 \\ z = 0 \end{cases}$

Find the first and second derivatives of the function f(x) = $x^{2}$ ( $\sqrt x - 5$ )

This question concerns the vector fieldF = −xy2 i − x2y j + z(x2 + y2) k.(a) Calculate the divergence of F. Use the divergence theorem to show thatthe flux of F over any closed surface is equal to zero. (b) A cylinder of height h and radius R has its base in the xy-plane and itsaxis of symmetry along the z-axis, as shown in the diagram below.Calculate the flux of F over the curved portion S of the surface of thecylinder by using the result of part (a) to relate the flux of F over S tothe flux of F over the flat top surface T and the flux of F over the flatbottom surface B.

Use the Divergence Theorem to calculate the surface integral F · dS, that is, calculate the flux of F across S.
$F (x, y, z) = x^{3} + c o s y i + y^{3} s i n x z j + z^{3} + 2 e k$
S is the surface of the solid bounded by the cylinder $y^{2} + z^{2} = 4$ and the planes $x = 0$ and $x = 4$ .

Addison Gross 2022-01-25

How do you find the critical point and determine whether it is a local maximum, local minimum, or neither for $f (x, y) = x^{2} + 4 x + y^{2}$ ?

Arely Briggs 2022-01-25

I have the following multivariable function: $f (x, y) = x y^{2}$ , and I must prove it is continuous at $(1, 1)$ . I have come to: $| {({(x - 1)}^{2} + {(y - 1)}^{2})}^{\frac{3}{2}} - 1 | \leq {({(x - 1)}^{2} + {(y - 1)}^{2})}^{\frac{3}{2}} + 1$ (Triangle inequality) $< δ^{3} + 1 = ϵ$ .
What can I do with this 1? Any tips are welcome and thank you in advance!

Ahmed Stewart 2022-01-25

I have a multivariable function that is composed of six variables; four of these are constants $C_{1}, C_{2}, C_{3}, C_{4}$ and the other two variables are x and y. The function is defined as follows
$X i (x, y, C_{1}, C_{2}, C_{3}, C_{4})$ I would like to find the maximum value of $X i$ over $x \in {- \infty, \infty}$ and $y \in {- \infty, \infty}$ . What is the rigorous mathematical notation I should use to describe this. Is this fine:
$max {X i (x, y, C_{1}, C_{2}, C_{3}, C_{4})}$

poveli1e 2022-01-25

$f (x, y) = x \sin (x) + \cos (x) - y \sin (x) + \frac{y^{2}}{2}$
My task is to find the critical points of this multivariable function (Determine the set of critical points of the function). Now I found the partial derivative to be $f_{x}^{'} = (x - y) \cos (x)$
$f_{y}^{'} = y - \sin (x)$
Wolfram Alpha says the critical point(s) is/are at $x = y$ ? How do I get the above in that form? The best I could do was get it in the form $x - y \sec (x) + \tan (x) = y$

Fallbasiss4 2022-01-25

Write a multivariable function (i.e z=f(x,y)) for a linear function that contains the points (−1,0),(0,2),(1,−1). Ive

kuntungw3 2022-01-24

Suppose that $x, y, z \in R$ with $a > 0$ a constant. Then, is the following true?
$f (x, y + c z, z) - f (x, y, z) = c \int_{0}^{1} (\partial_{x} + z \partial_{y}) f (x, y + θ z, z) d θ .$
It looks like the mean value theorem for multivariable function but I don't know why $\partial_{x}$ appears.

As you start exploring calculus and analysis, you will encounter multivariable calculus equations that are self-explanatory as well because all of them will contain at least two questions related to each variable being involved. See our multivariable calculus examples to receive more help and information regarding how these are used. The answers to solving these multivariable calculus questions should be based on finding the deterministic behavior. These are used in engineering and those fields where the parametric equations solver will provide you an optimal control of time dynamic systems. As an interesting subject, applying at least one equation in practice will keep you inspired!

Multivariable calculus

Solve for w in the equation y-1=4wy-6.

Solve for w: 0=-6u-7w-6.