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Multivariable Calculus

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Solving Differential Equations Using an Integrating Factor

A clever method for solving differential equations (DEs) is in the form of a linear first-order equation. This method involves multiplying the entire equation by an [more…]

Drawing with 3-D Cartesian Coordinates

The three-dimensional (3-D) Cartesian coordinate system (also called 3-D rectangular coordinates) is the natural extension of the 2-D Cartesian graph. The key difference is the addition of a third axis [more…]

Measuring Volume Under a Surface Using a Double Integral

A double integral allows you to measure the volume under a surface as bounded by a rectangle. Definite integrals provide a reliable way to measure the signed area between a function and the [more…]

Evaluating Double Integrals

Double integrals are usually definite integrals, so evaluating them results in a real number. Evaluating double integrals is similar to evaluating nested functions: You work from the inside out. [more…]

Evaluating Triple Integrals

Triple integrals are usually definite integrals, so evaluating them results in a real number. Evaluating triple integrals is similar to evaluating nested functions: You work from the inside out. [more…]

Identifying Ordinary, Partial, and Linear Differential Equations

Differential equations (DEs) come in many varieties. And different varieties of DEs can be solved using different methods. You can classify DEs as ordinary and partial Des. In addition to this distinction [more…]

Checking Differential Equation Solutions

Even if you don’t know how to find a solution to a differential equation, you can always check whether a proposed solution works. This is simply a matter of plugging the proposed value of the dependent [more…]

Solving Separable Differential Equations

Differential equations become harder to solve the more entangled they become. In certain cases, however, an equation that looks all tangled up is actually easy to tease apart. Equations of this kind are [more…]

Calculating Magnitude with Vectors

Vectors are commonly used to model forces such as wind, sea current, gravity, and electromagnetism. Calculating the magnitude of vectors is essential for all sorts of problems where forces collide. [more…]

Using Scalar Multiplication with Vectors

Multiplying a vector by a scalar is called scalar multiplication. To perform scalar multiplication, you need to multiply the scalar by each component of the vector. [more…]

Finding the Unit Vector of a Vector

Every nonzero vector has a corresponding unit vector,which has the same direction as that vector but a magnitude of 1. To find the unit vector u of the vector [more…]

Adding and Subtracting Vectors

You can add and subtract vectors on a graph by beginning one vector at the endpoint of another vector. You add and subtract vectors component by component, as follows: [more…]

How to Plot Cylindrical Coordinates

Cylindrical coordinates are simply polar coordinates with the addition of a vertical z-axis extending from the origin. While a polar coordinate pair is of the form [more…]

How to Plot Spherical Coordinates

Spherical coordinates are used — with slight variation — to measure latitude, longitude, and altitude on the most important sphere of them all, the planet Earth. Every point in space is assigned a set [more…]

How to Use a Partial Derivative to Measure a Slope in Three Dimensions

You can use a partial derivative to measure a rate of change in a coordinate direction in three dimensions. To do this, you visualize a function of two variables [more…]

Understanding Systems of Inequalities

In a system of inequalities, you see more than one inequality with more than one variable. Before pre-calculus, teachers tend to focus mostly on systems of linear inequalities. The graphs of those inequalities [more…]

How to Apply Basic Operations to Matrices

When you apply basic operations to matrices, it works a lot like operating on multiple terms within parentheses; you just have more terms in the "parentheses [more…]

How to Write a System in Matrix Form

In a system of linear equations, where each equation is in the form Ax + By + Cz + . . . = K, you can represent the coefficients of this system in matrix, called the [more…]

Writing a Matrix in Augmented Form

An alternative to writing a system of equations as the product of a coefficient matrix and variable matrix equaling an answer matrix is what's known as [more…]

How to Solve a System of Equations Using the Inverse of a Matrix

If you have a coefficient tied to a variable on one side of a matrix equation, you can multiply by the coefficient's inverse to make that coefficient go away and leave you with just the variable. For example [more…]

How to Solve Linear Systems that Have More than Two Equations

When your pre-calculus instructor asks you to solve larger systems of linear equations, these equations will involve more than two equations that go along with more than two variables. You can write these [more…]

How to Use Gaussian Elimination to Solve Systems of Equations

Gaussian elimination is probably the best method for solving systems of equations if you don't have a graphing calculator or computer program to help you. [more…]

How to Multiply Matrices by Each Other

Multiplying matrices is very useful when solving systems of equations. This is because you can multiply a matrix by its inverse on both sides of the equal sign to eventually get the variable matrix on [more…]

Decompose Partial Fractions in 8 Steps

When your pre-calculus instructor asks you to decompose partial fractions, it's actually not as messy as it sounds. The process of decomposing a partial fraction requires you to separate the fraction into [more…]

Writing a Matrix in Reduced Row Echelon Form

You can find the reduced row echelon form of a matrix to find the solutions to a system of equations. Although this process is complicated, putting a matrix into reduced row echelon form is beneficial [more…]

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