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6.5E: Exercises
https://math.libretexts.org/Courses/Mount_Royal_University/MATH_3200%3A_Mathematical_Methods/6%3A__Differentiation_of_Functions_of_Several_Variables/6.5%3A_The_Chain_Rule_for_Multivariable_Functions/6.5E%3A_Exercises
Suppose at a given time the $\displaystyle x$ resistance is $\displaystyle 100Ω$ , the $\displaystyle y$ resistance is $\displaystyle 200Ω,$ and the $\displaystyle z$ resistance is \(\display...Suppose at a given time the $\displaystyle x$ resistance is $\displaystyle 100Ω$ , the $\displaystyle y$ resistance is $\displaystyle 200Ω,$ and the $\displaystyle z$ resistance is $\displaystyle 300Ω.$ Also, suppose the $\displaystyle x$ resistance is changing at a rate of $\displaystyle 2Ω/min,$ the $\displaystyle y$ resistance is changing at the rate of $\displaystyle 1Ω/min$ , and the $\displaystyle z$ resistance has no change.
7.4: Triple Integrals
https://math.libretexts.org/Courses/Mount_Royal_University/MATH_3200%3A_Mathematical_Methods/7%3A_Multiple_Integration/7.4%3A_Triple_Integrals
\[\begin{align*} &\int_{x=0}^{x=1} \int_{y=0}^{y=x^2} \int_{z=0}^{z=y^2} xyz \, dz \, dy \, dx \\ &= \int_{x=0}^{x=1} \int_{y=0}^{y=x^2} \left. \left[xy \dfrac{z^2}{2} \right|_{z=0}^{z=y^2} \right] \,... $\begin{align*} &\int_{x=0}^{x=1} \int_{y=0}^{y=x^2} \int_{z=0}^{z=y^2} xyz \, dz \, dy \, dx \\ &= \int_{x=0}^{x=1} \int_{y=0}^{y=x^2} \left. \left[xy \dfrac{z^2}{2} \right|_{z=0}^{z=y^2} \right] \, dy \, dx = \int_{x=0}^{x=1} \int_{y=0}^{y=x^2} \left( x \dfrac{y^5}{2}\right) dy \, dx = \int_{x=0}^{x=1} \left. \left[ x\dfrac{y^6}{12} \right|_{y=0}^{y=x^2}\right] dx = \int_{x=0}^{x=1} \dfrac{x^{13}}{12} dx = \left. \dfrac{x^{14}}{168}\right|_{x=0}^{x=1} = \dfrac{1}{168}, \end{align*}$
8: Partial Differential Equations
https://math.libretexts.org/Courses/Mount_Royal_University/MATH_3200%3A_Mathematical_Methods/8%3A_Partial_Differential_Equations
A partial differential equation is an equation that involves an unknown function of more than one independent variable and one or more of its partial derivatives. Many important and interesting physic...A partial differential equation is an equation that involves an unknown function of more than one independent variable and one or more of its partial derivatives. Many important and interesting physical phenomena are modelled by the functions of several variables that satisfy certain partial differential equations. In this chapter, we will learn a few particular partial differential equations that arise in physical sciences such as heat equation, Laplace equation etc.
MATH 3200: Mathematical Methods
https://math.libretexts.org/Courses/Mount_Royal_University/MATH_3200%3A_Mathematical_Methods
Mathematical Methods provides an introduction to vector calculus, ordinary differential equations, and partial differential equations including a variety of applications. Topics include: optimization,...Mathematical Methods provides an introduction to vector calculus, ordinary differential equations, and partial differential equations including a variety of applications. Topics include: optimization, line and surface integrals, Green’s theorem, Stokes’ theorem, the Divergence theorem, and the theory of systems of linear differential equations.
5.4: Motion in Space
https://math.libretexts.org/Courses/Mount_Royal_University/MATH_3200%3A_Mathematical_Methods/5%3A_Vector-Valued_Functions/5.4%3A_Motion_in_Space
The effect of gravity is in a downward direction, so Newton’s second law tells us that the force on the object resulting from gravity is equal to the mass of the object times the acceleration resultin...The effect of gravity is in a downward direction, so Newton’s second law tells us that the force on the object resulting from gravity is equal to the mass of the object times the acceleration resulting from gravity, or $\vec F_g=m\vec a$ , where $\vec F_g$ represents the force from gravity and $\vec a = -g\,\hat{\mathbf j}$ represents the acceleration resulting from gravity at Earth’s surface.
9.1E: Exercises
https://math.libretexts.org/Courses/Mount_Royal_University/MATH_3200%3A_Mathematical_Methods/9%3A_Vector_Calculus/9.1%3A_Vector_Fields/9.1E%3A_Exercises
For the following exercises, let $\vecs{F}=x\, \hat{\mathbf i}+y\, \hat{\mathbf j}, \vecs{G}=−y\, \hat{\mathbf i}+x\, \hat{\mathbf j}, and \vecs{H}=−x\, \hat{\mathbf i}+y\, \hat{\mathbf j}.$ Match ...For the following exercises, let $\vecs{F}=x\, \hat{\mathbf i}+y\, \hat{\mathbf j}, \vecs{G}=−y\, \hat{\mathbf i}+x\, \hat{\mathbf j}, and \vecs{H}=−x\, \hat{\mathbf i}+y\, \hat{\mathbf j}.$ Match the vector fields with their graphs in (I)−(IV).
7.6E
https://math.libretexts.org/Courses/Mount_Royal_University/MATH_3200%3A_Mathematical_Methods/7%3A_Multiple_Integration/7.6%3A_Calculating_Centers_of_Mass_and_Moments_of_Inertia/7.6E
The solid $Q$ has the moment of inertia $I_x$ about the $yz$ -plane given by the triple integral \[\int_0^2 \int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}} \int_{\frac{1}{2}(x^2+y^2)}^{\sqrt{x^2+y^2}} (y^2 +...The solid $Q$ has the moment of inertia $I_x$ about the $yz$ -plane given by the triple integral $\int_0^2 \int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}} \int_{\frac{1}{2}(x^2+y^2)}^{\sqrt{x^2+y^2}} (y^2 + z^2)(x^2 + y^2) dz \space dx \space dy.$ Show that the moments of inertia $I_x, \space I_y$ , and $I_z$ about the $yz$ -plane, $xz$ -plane, and $xy$ -plane, respectively, of the unit ball centered at the origin whose density is $\rho (x,y,z) = e^{-x^2-y^2-z^2}$ are the same.
5: Vector-Valued Functions
https://math.libretexts.org/Courses/Mount_Royal_University/MATH_3200%3A_Mathematical_Methods/5%3A_Vector-Valued_Functions
He stated that comets that had appeared in 1531, 1607, and 1682 were actually the same comet and that it would reappear in 1758. Halley’s Comet follows an elliptical path through the solar system, wit...He stated that comets that had appeared in 1531, 1607, and 1682 were actually the same comet and that it would reappear in 1758. Halley’s Comet follows an elliptical path through the solar system, with the Sun appearing at one focus of the ellipse. Kepler’s third law of planetary motion can be used with the calculus of vector-valued functions to find the average distance of Halley’s Comet from the Sun.
1.4E: Exercises
https://math.libretexts.org/Courses/Mount_Royal_University/MATH_3200%3A_Mathematical_Methods/1%3A_Power_Series/1.4%3A_Working_with_Taylor_Series/1.4E%3A_Exercises
$\displaystyle y''=\sum_{n=0}^∞(n+2)(n+1)a_{n+2}x^n$ and $\displaystyle y′=\sum_{n=0}^∞(n+1)a_{n+1}x^n$ so $\displaystyle y''−y′+y=0$ implies that \(\displaystyle (n+2)(n+1)a_{n+2}−(n+1)a_{n+1}+... $\displaystyle y''=\sum_{n=0}^∞(n+2)(n+1)a_{n+2}x^n$ and $\displaystyle y′=\sum_{n=0}^∞(n+1)a_{n+1}x^n$ so $\displaystyle y''−y′+y=0$ implies that $\displaystyle (n+2)(n+1)a_{n+2}−(n+1)a_{n+1}+a_n=0$ or $\displaystyle a_n=\frac{a_{n−1}}{n}−\frac{a_{n−2}}{n(n−1)}$ for all $\displaystyle n⋅y(0)=a_0=1$ and $\displaystyle y′(0)=a_1=0,$ so $\displaystyle a_2=\frac{1}{2},a_3=\frac{1}{6},a_4=0$ , and $\displaystyle a_5=−\frac{1}{120}$ .
6E: Chapter Review Excersies
https://math.libretexts.org/Courses/Mount_Royal_University/MATH_3200%3A_Mathematical_Methods/6%3A__Differentiation_of_Functions_of_Several_Variables/6.8%3A_Lagrange_Multipliers/6.8E%3A/6E%3A_Chapter_Review_Excersies
1) Find the equations of the tangent plane and normal line to the graph of \[f(x,y)=\tan^{-1}(y/x)\[ at $(1,-1).$ The temperature at position $(x,y)$ in a region of the $xy-$ plane is \(T^{\circ}...1) Find the equations of the tangent plane and normal line to the graph of \[f(x,y)=\tan^{-1}(y/x)\[ at $(1,-1).$ The temperature at position $(x,y)$ in a region of the $xy-$ plane is $T^{\circ} C,$ where $T(x,y)= x^2 -x+y+2y^2.$ The material used for the bottom and front of the box is five times as costly (per square metre) as the material used for the back and the other two sides. what should be the dimensions of the box to minimize the cost of materials?
Summary Table Of Integrals
https://math.libretexts.org/Courses/Mount_Royal_University/MATH_3200%3A_Mathematical_Methods/Summary_Tables/Summary_Table_Of_Integrals
\int \sec u\; du&=\ln|\sec u+\tan u|+c \\ &\\ \int \csc( {u})du &= \ln|\csc( {u}) - \cot( {u})| + c\\ &\\ \int e^{\lambda u}\cos\omega u\; du&={e^{\lambda u}(\lambda\cos\omega u+\omega\sin\omega u)\ov...\int \sec u\; du&=\ln|\sec u+\tan u|+c \\ &\\ \int \csc( {u})du &= \ln|\csc( {u}) - \cot( {u})| + c\\ &\\ \int e^{\lambda u}\cos\omega u\; du&={e^{\lambda u}(\lambda\cos\omega u+\omega\sin\omega u)\over \lambda^2+\omega^2}+c \\ &\\ If $f(x)$ is continuous over $[a,b]$ except at a point $c$ in $(a,b)$ , then $\int ^b_af(x)dx=\int ^c_af(x)dx+\int ^b_cf(x)dx,$ provided both $\int ^c_af(x)dx$ and $\int ^b_cf(x)dx$ converge.

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