18 Power Packed Formula to Boost Your Math Exam Grades
Every college student aspires to excel in mathematics, which demands adequate preparation. While studying before you sit for your exam is the first step to scoring highly, you must apply the right formulae when solving questions. We have different formulae for solving problems in various topics, starting from simple to complex tasks.
Some formulae are applicable across topics. Derivatives in calculus are applied in vectors and mechanics when finding velocity and acceleration. Mastering the formulae in one area can help solve problems in another. In this article, we have listed the common formulae to boost your math exam grades.
Common Formulae and Theorems in Vectors
With 40+ vector calculus courses at the college level, you will do a minimum of four vector courses depending on your diploma or degree course. Here are the fundamental theorems you must know before sitting for your exam.
- Rank-Nullity theorem: Rank(A) + dim(Nul(A)) = n, where A is m × n. A vector A is null if Ax=0.
- Cauchy-Schwarz: |u · v| ≤ ||u|| ||v||. The dot product of two vectors, u and v, is equivalent to the product of their magnitude.
- The gradient theorem: ∫Δf.ds = f(h)-f(k), where k and h are the limits of integration.
- Green’s theorem: ∫CF. ds= ∫∫D (∂F2/∂x _∂F1/∂y)dA. This theorem connects a double integral to a region over a line integral with a boundary.
- Stokes theorem: ∫CF. ds= ∫∫S curl F.ds Where C is the boundary to a surface S.
- The divergence theorem: ∫∫S F. ds = ∫∫∫Y div Fdv where s is a normal vector pointing to the exterior of the vector field, and y is the total fluid expansion.
In addition to the above theorems, you should master the following formulae.
- Linear transformation: T(u + v) = T(u) + T(v), note that T(cu) = cT(u), where c is any number.
- Linearly independence: vectors v1+v2+v3…… are linearly independent if none of them can be expressed as a linear combination of others. a1v1 + a2v2 + · · · + anvn = 0 ⇒ a1 = a2 = · · · = an = 0.
- Linearly dependence: Linearly dependent vectors have non-zero solutions, that is, a1v1+a2v2+a3v3……+anvn ≠ 0.
In addition to the common operations on matrices, the following theorems and formulae are essential.
- The eigenvalues from any Eigen spaces are orthogonal for any symmetric matrix. Note that a symmetric matrix contains eigenvalues diagonalizable orthogonally and is denoted as [A =AT].
- You can write a matrix from a quadratic equation by writing the x2 -coefficients on the diagonal and then distributing the other terms. For example, if the x1x2 term is 6, then the (1, 2)th and (2, 1)th entry of A is 3. Then orthogonally diagonalize A =AT. The quadratic form becomes λ1y2 + · · · + λnyn, where λi are the eigenvalues.
- The invertible matrix theorem. Finding singular solutions to matrices requires knowledge of the inverse and how it helps you solve for unknown elements. Given a 2x2 matrix a b c d , the determinant is obtained as ad-bc while the inverse is
1/ad-bc d -b -c a . The inverse of a matrix A is denoted as A-1.
- Diagonalizability. A matrix is diagonalizable if it has n eigenvalues with more than one linearly independent eigenvector. The matrix is diagonalizable if the quadratic formed from the eigenvalues has more than one root ≠ 0.
Theorem: IF A has n distinct eigenvalues, THEN A is diagonalizable, but the opposite is not always true. Note that a matrix can be diagonalizable even if it’s not invertible.
How to diagonalize: To find the eigenvalues, calculate det(A − λI) and find the roots. To find the eigenvectors, for each λ, find a basis for Nul(A − λI), which you do by row-reduction or zeroing.
- Rational Roots: For a given polynomial to have rational roots, the leading coefficient must be a multiple of the fraction's denominator, while the constant term must be divisible by the numerator.
Rational roots theorem: If p(λ) = 0 has a rational root r = a b, then a divides the constant term of p, and b divides the leading coefficient.
In addition to the general methods of solving differential equations, the following theorems and formulae are crucial.
Homogeneous solutions: When solving an auxiliary equation, replace the equation with a polynomial so that y111 becomes r3 etc. Then find the zeros using the rational roots theorem and long division.
Note that Simple zeros’ give you ert, Repeated zeros (multiplicity m) give you Aert + Btert + · · · Ztm-1ert, Complex zeros r = a + bi give you Aeat cos(bt) + Beat sin(bt).
For non-homogeneous differential equations, you can use undetermined coefficients to solve them.
- y(t) = y0(t) + yp(t) in homogeneous equations changes to
yp = t s (Amtm · · · + A1t + 1)ert, where if r is a root of the auxiliary equation with multiplicity m, then s = m, and if r is not a root, then s = 0. If the non-homogeneous term is Ctmeat sin(βt), then: yp = t s (A · · · + A1t + 1)e at cos(βt) + t s (Bmtm · · · + B1t + 1)e rt sin(βt), where s = m.
Solution by variation of parameters. Before using this method, the leading coefficient should be 1. If it is more than one, divide all the differential equation coefficients by the leading and use the matrix method to solve. Suppose yp(t) = v1(t)y1(t) + v2(t)y2(t), where y1 and y2 are your homogeneous solutions, then, y1 y2 v1 =0. y1 y2 y11 y21 v11 v21 =0 f(t)
To solve for t, find the inverse and solve before integrating to find v.
Other Important for Solving Differential Equations
When solving differential equations involving trigonometric ratios, the following formulae are essential.
R sec(t) = ln |sec(t) + tan(t)|,
R tan(t) = ln |sec(t)|,
R tan2 (t) = tan(x) − x,
R ln(t) = tln(t) − t
Linear independence: f, g, h are linearly independent if af(t) + bg(t) + ch(t) = 0 ⇒ a = b = c = 0. You can prove that an equation is linearly dependent directly. For linear independence, you should use the Wronskian.
Wf(t) = f(t) g(t) fI(t) fgI(t) (for 2 functions).
If your differential equation has three functions,
Wf(t) = f(t) g(t) h(t) fI(t) gI(t) hI(t) fII(t) gII(t) hII(t)
Systems Differential Equations
A matrix whose columns are independent is the solution to the system's equation. For a matrix A, To solve x| = Ax:
x(t) = Aeλ1tv1 + Beλ2tv2 + e λ3tv3 (λi are your eigenvalues, vi are your eigenvectors).
Complex eigenvalues: If λ = α + iβ, and v = a + ib. Then: x(t) = A e αt cos(βt)a − e αt sin(βt)b + B e αt sin(βt)a + e αt cos(βt)b
In the above equation, you only need to consider one complex eigenvalue.
Also, 1 a+bi = a−bi a2+b 2
Generalized eigenvectors: You only need to obtain one solution by using the following equation to solve for u: (A − λI)(u) = v.
Partial Differential Equations
When solving partial differential equations, you should make them solvable by changing them to ordinary differential equations. Different series to help you solve PDEs include:
- The Fourier series:
f(x) = A0 +⅀∞n=1 Ancosnx +⅀∞n=1 Bnsinnx . This formula expands a function f(x) with an infinite number of sines and cosines.
- The cosine series: A function defined on (O, T), f(x)≅ ⅀∞n=1 bmcos(ℼmx/T) where:
a0 = 1/2T∫T-T f(x)dx
am = 2/T∫0Tf(x)cosx(ℼmx/T) . this series decomposes PDEs and makes them solvable.
- The sine series: A function defined on (O, T), f(x)≅ ⅀∞n=1 bmsin(ℼmx/T)
a0 = 1/2T∫T-T f(x)dx
am = 2/T∫0Tf(x)sinx(ℼmx/T)
- Tabular integration is used to integrate polynomials by listing the values on a table and finding the functions' derivatives until you obtain zero.
You Can Do More…
The above theorems and formulae are commonly used in mathematics and are applicable across different topics. However, the best way to grasp them is by practicing and solving more problems. It is almost impossible to master the formulae overnight, but consistent study and application will help you memorize them quickly.
If you have problems grasping concepts in your math assignment, you can find help from online experts. They are here to guide you through your notes and assist you in solving your homework questions. This team of experienced assignment helpers has a proven record of simplifying and explaining technical concepts in simple terms to enable you to write your homework and exams perfectly.