Multivariable Calculus Examples: Learn Complex Calculus Problems

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Calculus and Analysis
Multivariable calculus

Recent questions in Multivariable calculus

Burhan Hopper 2021-01-05

A concert promoter produces two kinds of souvenir shirt, one kind sells for $18 ad the other for $25. The company determines, the total cost, in thousands of dollars, of producting x thousand of the $18 shirt and y thousand of the $25 shirt is given by
$C (x, y) = 4 x^{2} - 6 x y + 3 y^{2} + 20 x + 19 y - 12 .$
How many of each type of shirt must be produced and sold in order to maximize profit?

Maiclubk 2021-01-05

Let $z = e^{x^{2} + 3 y} + x^{3} y^{2}$ , where $x = t \cos r and y = r t^{4}$ . Use the chain rule fot multivariable functions (Cals III Chain Rule) to find $\frac{\partial z}{\partial r}$ and $\frac{\partial z}{\partial} t)$ . Give your answers in terms of r and t only. Be sure to show all of your work.

Braxton Pugh 2021-01-04

Suppose S is a region in the xy-plane with a boundary oriented counterclockwise. What is the normal to S? Explain why Stokes’ Theorem becomes the circulation form of Green’s Theorem.

Maiclubk 2021-01-02

Consider the vector field $F = ⟨ 5 z, x, 5 y ⟩$ and the surface which is the part of the elliptic paraboloid $z = x^{2} + 5 y^{2}$ that lies below the plane z = 5. Calculate curl(F) and then apply Stokes' Theorem to compute the exact magnitude of the flux of curl(F) through the surface using line integral. You do not need to cinfirm your answer by evaluating the double integral of curl(F) over the surface(the right-hand side of Stokes' Theorem).

slaggingV 2021-01-02

By considering different path of approach, show that the function $f (x, y) = \frac{x^{2} + y}{y}$ has no limis $(x, y) \to (0, 0)$

generals336 2020-12-29

Use the Divergence Theorem to find the flux of $F = x y^{2} i + x^{2} y j + y k$ outward through the surface of the region enclosed by the cylinder $x^{2} + y^{2} = 1$ and the planes z = 1 and z =-1.

UkusakazaL 2020-12-17

A surface is represented by the following multivariable function,
$f (x, y) = \frac{1}{3} x^{3} + y^{2} - 2 x y - 6 x - 3 y + 4$
a) Calculate $f_{x x}, f_{y x}, f_{x y} and f_{y y}$
b) Calculate coordinates of stationary points.
c) Classify all stationary points.

Ramsey 2020-12-17

Let $F (x, y) = ⟨ x y^{2} + 8 x, x^{2} y - 8 y ⟩$ . Compute the flux $\oint F \cdot n d s$ of F across a simple closed curve that is the boundary of the half-disk given by $x^{2} + y^{2} \leq 7, y \geq 0$ using the vector form Greens

OlmekinjP 2020-12-16

Let $f = [x^{2} y^{2}, - \frac{x}{y^{2}}]$ and $R : 1 \leq x^{2} + y^{2}, + 4, x \geq 0, y \geq x$ . Evaluate $\int_{C} F (r) \cdot d r$ counterclockwise around the boundary C of the region R by Green's theorem.

Reeves 2020-12-13

Use Green's Theorem to evaluate the line integral.
$(2 x + \ln 2) d y - (8 y^{2} + sinhx) d x$

necessaryh 2020-12-06

Use the Divergence Theorem to calculate the surface integral $\int \int_{S} F \cdot d S$ , that is, calculate the flux of F across S.
$F (x, y, z) = (\cos (z) + x y^{2}) i + x e^{- z} j + (\sin (y) + x^{2} z) k$
S is the surface of the solid bounded by the paraboloid $z = x^{2} + y^{2}$ and the plane z = 9.

foass77W 2020-12-03

Use Green's Theorem to evaluate the line integral along the given positively oriented curve.
$\int_{C} x y^{2} d x + 4 x^{2} y d y$
$C$ is the triangle with vertices $(0, 0), (3, 3)$ , and $(3, 6)$

geduiwelh 2020-12-02

Green’s Theorem, flux form Consider the following regions R and vector fields F.
a. Compute the two-dimensional divergence of the vector field.
b. Evaluate both integrals in Green’s Theorem and check for consistency.
$F = ⟨ x, y ⟩, R = {(x, y) : x^{2} + y^{2} \leq 4}$

pancha3 2020-12-02

Evaluate the line integral $\oint_{C} x y d x + x^{2} d y$ , where C is the path going counterclockwise around the boundary of the rectangle with corners (0,0),(2,0),(2,3), and (0,3). You can evaluate directly or use Greens

Harlen Pritchard 2020-11-27

Comsider this, multivariable fucntion
$f (x, y) = 3 x y - x^{2} + 5 y^{2} - 25$
a)What is the value of f(-1,3)?
b)Find all x-values such that f(x,x)=0

cistG 2020-11-23

Write formulas for the indicated partial derivatives for the multivariable function.
$k (a, b) = 3 a b^{4} + 8 ({1.4}^{b})$
a) $\frac{\partial k}{\partial a}$
b) $\frac{\partial k}{\partial b}$
c) $\frac{\partial k}{\partial b} ∣_{a = 3}$

coexpennan 2020-11-20

Consider this multivariable function. $f (x, y) = y e^{3 x} + y^{2}$
a) Find $f_{y} (x, y)$
b) What is value of $f_{\times} (0, 3)$ ?

Jaya Legge 2020-11-08

Use Green's Theorem to find $\int_{C} \vec{F} \cdot d \vec{r}$ where $\vec{F} = ⟨ y^{3}, - x^{3} ⟩$ and C is the circle $x^{2} + y^{2} = 3$ .

djeljenike 2020-11-08

Divergence Theorem for more general regions Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D.
$F = ⟨ z - x, x - y, 2 y - z ⟩$ , D is the region between the spheres of radius 2 and 4 centered at the origin.

illusiia 2020-11-02

Suppose that the plane region D, its boundary curve C, and the functions P and Q satisfy the hypothesis of Greens

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!

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