Calculus 3 | mathXplain
 
5 topics, 51 short and super clear lessons

This Calculus 3 course includes 51 short and super clear lessons that take you through 5 topics and help you navigate the bumpy roads of Calculus 3. The casual style makes you feel like you are discussing some simple issue, such as cooking scrambled eggs.

Table of contents: 

The course consists of 5 sections: Matrices and vectors, Determinants, eigenvectors and eigenvalues, Functions of two variables, Double integrals, Differential equations

MATRICES

  • Matrices - Matrices are really harmless creatures in mathematics. An nXk matrix is simply a rectangular array of numbers, arranged in n rows and k columns.
  • Matrix operations - Scalar multiplication, addition and multiplication.
  • Square and diagonal matrices - It is a square-shaped matrix with the same number of rows and columns.The diagonal matrix is a square matrix where all elements outside the main diagonal are zero.
  • Transpose - The transpose matrix is created by swapping the rows and the columns of the matrix

 

VECTORS

  • Dot product - The dot product, or sometimes inner product, is an algebraic operation that takes two vectors and turns into a single number.
  • Cross product - The cross product, is an algebraic operation that takes two vectors and turns into anather vector.
  • Dyadic product - The dyadic product, is an algebraic operation that takes two vectors and turns into a matrix.
  • Angle between the two vectors - To calculate the angle between the two vectors, we write their dot product using two formulas. 
  • The equation of the line - We will develop the various forms for the equation of lines in plane and three dimensional space.
  • The equation of the plane - Here we will develop the equation of a plane.

 

THE DETERMINANT

  • Definition of determinant - The determinant takes a matrix, and turns it into a single number. Let’s see how that happens.
  • Sarrus' Rule - There is a rule for calculating the determinant of a 3x3 matrix. It is known as Sarrus' rule. 
  • The expansion rule - The essence of the expansion rule is that for an nxn matrix of any size, the rather nasty calculation of its determinant can be reduced to calculating the determinants of 2x2 matrices, which is fairly easy.
  • Singular and invertible matrices - The nxn matrices can be classified into two large groups. There are those whose determinant is zero, and there are those whose is non-zero. 
  • Eigenvector - An eigenvector of an nxn matrix A is a non-zero vector, where there is a real number such that A multiplied by the vector is equal to the number multiplied by the vector.
  • Eigenvalue - An eigenvalue of an nxn matrix A is a number, where there is a non-zero vector such that A multiplied by the vector is equal to the number multiplied by the vector.
  • Characteristic equation - The solutions to the characteristic equation will be the eigenvalues. 
  • The diagonal form - If an nxn matrix has n independent eigenvectors, then the matrix has a diagonal form, where the main diagonal contains the eigenvalues, and all other elements are zero.
  • Definiteness of matrices - In order to figure out definiteness, we will need these leading principal minors – more precisely, we will need their signs. 
  • Quadratic forms - The quadratic form a homogeneous quadratic (second-degree) polynomial. This means the x variables are either raised to the second power, or raised to the first power but multiplied by another x, and that counts as quadratic as well.
  • Definitness of quadratic forms - The matrix of the quadratic form helps us determine the definiteness. 
 

DIFFERENTATION OF TWO VARIABLE FUNCTIONS

  • Two-variable functions - Functions of two variables take two real numbers and assign a third real number to them.
  • Partial derivatives - Just like in the one-variable case, we will have to differentiate here, too, but no we have x as well as y, so we have to differentiate with respect to x and also with respect to y, which should be twice as much fun.
  • Second order derivatives - Both first order partial derivatives can be further differentiated with respect to x as well as y, too. This way we get four second order derivatives.
  • Stationary points - At stationary points the function can have a minimum, a maximum or a saddle point.
  • Hessian matrix - If the Hessian matrix is negative definite, then there is a maximum point, if it’s positive definite, then there is a minimum point, and if it’s indefinite, then there is a saddle point.
  • The gradient - The vector made up of the of the function's partial derivatives with respect to x and y is called the gradient of the function.
  • The tangent plane - The tangent of a single variable function is a line, and the tangent of a two-variable function is a plane.
  • Directional derivative - It means that there is a mountain climber standing at point P on the surface, and decides to move in v direction. The directional derivative tells him how steep he would have to climb.
  • The implicit differentation rule - If the function has an explicit form, it was unnecessary to suffer through the implicit differentiation, but, if cannot be expressed in any way, so we are forced to use implicit differentiation.

DOUBLE INTEGRALS

  • Normal region - The region is from a to b on the x axis, and from c to d on the y axis.
  • Double integrals - Double integrals can be used to compute volumes under various surfaces.
  • Polar coordinates - The idea of polar coordinates is that we replace the x and y coordinates with new ones.

DIFFERENTIAL EQUATIONS

  • Differential equations - Differential equations are equations where the unknowns are functions. The equations contain various derivatives and exponentials of these functions.
  • Order of equations - It indicates the highest derivative of the function in the equation.
  • Linearity - If the equation includes the unknown function and its derivatives only in first-degree form, then the equation is linear.
  • Ordinary differential equation - If the differential equation contains a function of one variable, then it is called an ordinary differential equation (ODE);
  • Separable differential equations - Move all y terms to the dy side, and move all x terms to the dx side, and integrate both sides to get the solution.
  • General solution - We get the general solution if we give C a arbitrary value.
  • Particular solution - We get the particular solution if we give C a fix value.
  • Exact differential equations - The p(x,y)dx+q(x,y)dy=0 equation is exact, if there exists a function F(x,y) such that F’x=p and F’y=q, then, the solution for the equation is that function F(x,y)=C
  • Integrating factor - If the equation is not exact, then we try to make it exact by using an integrating factor.
  • First-order linear differential equations - In general, a first-order linear differential equation contains a y’ and it also contains a first-order y.
  • First-order constatnt coeffitient linear differential equations - This type is a special case of the first-order linear differential equations.
  • First order homogeneous equation - The homogeneous equation is y’+ay=0
  • Homogeneous solution - The homogeneous solution is the solution of homogeneous equation.
  • Particular solution - We can obtain the particular solution based on the function on the right side, using a very funny procedure called “Method of Undetermined Coefficients”, or “Trial Functions Method”.
  • Trial Functions Method - We can obtain the particular solution based on the function on the right side, using a very funny procedure called “Method of Undetermined Coefficients”, or “Trial Functions Method”.
  • Method of Undetermined Coeffitients - We can obtain the particular solution based on the function on the right side, using a very funny procedure called “Method of Undetermined Coefficients”, or “Trial Functions Method”.
  • Resonance in first order cases - This occurs when there is eAx in the particular solution, and its exponent is exactly the same as the exponent of the homogeneous solution.
  • Second order homogeneous equation - The homogeneous equation is ay"+by'+cy=0
  • Homogeneous solution - The homogeneous solution is the solution of homogeneous equation.
  • Characteristic equation - In order to solve the homogeneous equation, we have to solve this quadratic equation.
  • Resonance in second order cases - One term of the homogeneous solution is equal to a term of the particular solution. That means that sadly, there is resonance. When the characteristic equation has only one real solution, there could be dual resonance.
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