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- 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_ExercisesSuppose 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.
- 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] \,...\[\begin{align*} &\int_{x=0}^{x=1} \int_{y=0}^{y=x^2} \int_{z=0}^{z=y^2} xyz \, dz \, dy \, dx \\ &= \int_{x=0}^{x=1} \int_{y=0}^{y=x^2} \left. \left[xy \dfrac{z^2}{2} \right|_{z=0}^{z=y^2} \right] \, dy \, dx = \int_{x=0}^{x=1} \int_{y=0}^{y=x^2} \left( x \dfrac{y^5}{2}\right) dy \, dx = \int_{x=0}^{x=1} \left. \left[ x\dfrac{y^6}{12} \right|_{y=0}^{y=x^2}\right] dx = \int_{x=0}^{x=1} \dfrac{x^{13}}{12} dx = \left. \dfrac{x^{14}}{168}\right|_{x=0}^{x=1} = \dfrac{1}{168}, \end{align*}\]
- https://math.libretexts.org/Courses/Mount_Royal_University/MATH_3200%3A_Mathematical_Methods/8%3A_Partial_Differential_EquationsA 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.
- https://math.libretexts.org/Courses/Mount_Royal_University/MATH_2200%3A_Calculus_for_Scientists_II/Appendix/A1%3A_Introduction_to_Differential_Equations/3.5%3A_Existence_and_Uniqueness_of_Solutions_of_Nonlinear_EquationsLet \(y\) be any solution of Equation \ref{eq:3.5.12}. Because of the initial condition \(y(0)=-1\) and the continuity of \(y\), there’s an open interval \(I\) that contains \(x_0=0\) on which \(y\) h...Let \(y\) be any solution of Equation \ref{eq:3.5.12}. Because of the initial condition \(y(0)=-1\) and the continuity of \(y\), there’s an open interval \(I\) that contains \(x_0=0\) on which \(y\) has no zeros, and is consequently of the form Equation \ref{eq:3.5.11}. Setting \(x=0\) and \(y=-1\) in Equation \ref{eq:3.5.11} yields \(c=-1\), so
- https://math.libretexts.org/Courses/Las_Positas_College/Math_27%3A_Number_Systems_for_Educators/08%3A_Additional_Activities/8.05%3A_5._Egyptian_PizzaExplain to students how if that was the case then they would all get at least half a loaf, so you would use 4 of the pizzas to give all 8 of them half a pizza each. Although they had a notation for 1 ...Explain to students how if that was the case then they would all get at least half a loaf, so you would use 4 of the pizzas to give all 8 of them half a pizza each. Although they had a notation for 1 / 2 and 1 / 3 and 1 / 4 and so on (these are called reciprocals or unit fractions since they are 1 / n for some number n), their notation did not allow them to write 2 / 5 or 3 / 4 or 4 / 7 as we would today.
- https://math.libretexts.org/Courses/Mount_Royal_University/MATH_2200%3A_Calculus_for_Scientists_II/Summary_Tables
- https://math.libretexts.org/Courses/Mount_Royal_University/MATH_3200%3A_Mathematical_MethodsMathematical 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.
- https://math.libretexts.org/Courses/Mount_Royal_University/MATH_3200%3A_Mathematical_Methods/5%3A_Vector-Valued_Functions/5.4%3A_Motion_in_SpaceThe 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.
- https://math.libretexts.org/Courses/Mount_Royal_University/MATH_2200%3A_Calculus_for_Scientists_II/1%3A_Applications_of_IntegrationFrom geometric applications such as surface area and volume, to physical applications such as mass and work, to growth and decay models, definite integrals are a powerful tool to help us understand an...From geometric applications such as surface area and volume, to physical applications such as mass and work, to growth and decay models, definite integrals are a powerful tool to help us understand and model the world around us. Figure \(\PageIndex{1}\): Hoover Dam is one of the United States’ iconic landmarks, and provides irrigation and hydroelectric power for millions of people in the southwest United States. (credit: modification of work by Lynn Betts, Wikimedia).
- https://math.libretexts.org/Courses/Mount_Royal_University/MATH_2200%3A_Calculus_for_Scientists_II/Appendix/A1%3A_Introduction_to_Differential_Equations/3.7%3A_Exact_EquationsSince the solutions of Equation \ref{eq:3.8.2} and Equation \ref{eq:3.8.3} will often have to be left in implicit form we will say that \(F(x,y)=c\) is an implicit solution of Equation \ref{eq:3.8.1} ...Since the solutions of Equation \ref{eq:3.8.2} and Equation \ref{eq:3.8.3} will often have to be left in implicit form we will say that \(F(x,y)=c\) is an implicit solution of Equation \ref{eq:3.8.1} if every differentiable function \(y=y(x)\) that satisfies \(F(x,y)=c\) is a solution of Equation \ref{eq:3.8.2} and every differentiable function \(x=x(y)\) that satisfies \(F(x,y)=c\) is a solution of Equation \ref{eq:3.8.3}
- https://math.libretexts.org/Courses/Las_Positas_College/Math_27%3A_Number_Systems_for_Educators/08%3A_Additional_Activities/8.02%3A_2._Regular_TilingA polygon is a closed 2-dimensional figure with straight sides A regular n-gon is a polygon with exactly n sides, where all sides are of equal length and all interior angles of the polygon are equal. ...A polygon is a closed 2-dimensional figure with straight sides A regular n-gon is a polygon with exactly n sides, where all sides are of equal length and all interior angles of the polygon are equal. The sum of the interior angles of a regular n-gon is 180°(n - 2). A regular 3-gon is an equilateral triangle. A regular 4-gon is a square. A regular 5-gon is a regular pentagon. A regular 6-gon is a regular hexagon. A regular 7-gon is a regular heptagon. A regular 8-gon is a regular octagon.
