# MATH10121 Calculus And Vectors A Notes | Assignment Help | Syllabus

## Country: United Kingdom

### University: University Of Manchester

The course teaches the fundamental concepts of complex numbers and connects them to calculus’ conventional rational and transcendental functions. Limits, differentiation, and integration are reinterpreted as fundamental notions. The calculus is applied using techniques that have been established and reinforced. The calculus of functions of more than one variable, vector calculus, integration in the plane, Green’s theorem, Stokes’ theorem, and Gauss’ theorem are all introduced, followed by the calculus of functions of more than one variable, vector calculus, integration in the plane, Green’s theorem, Stokes’ theorem, and Gauss’ theorem.

Aims

Real and complex numbers, standard functions, curve sketching, Taylor series, limits, continuity, differentiation, integration, vectors in two and three dimensions, and the calculus of functions with more than one variable are all covered in this course unit.

the consequences of learning

Students who successfully complete this module will be eligible to:

Work with complex numbers to connect trigonometric, hyperbolic, and exponential functions.

Using Euler’s formula and de Moivre’s theorem, evaluate complex roots and represent complex numbers on the complex plane.

In Cartesian or polar coordinates, draw polynomial, rational, inverse, and certain conventional functions of a single variable.

Limits and derivatives of algebraic functions, including implicit and parametric functions, are evaluated and interpreted.

Choose and use techniques for assessing integrals of single-variable functions.

Construct and modify Taylor series of one- and two-variable scalar functions.

Use techniques like scalar and vector products to manipulate vectors and use them to solve geometric issues.

Construct, deploy, and analyze derivatives of multivariable scalar- and vector-valued functions.

Using techniques like the Jacobian and Green’s theorem, construct, evaluate, and understand basic integrals of two-variable functions.

For conservative and non-conservative vector fields, construct and evaluate line integrals.

Using discriminants and Lagrange multipliers, locate and categorize the extrema of two-variable functions.

Syllabus

1. Complex Numbers: Imaginary numbers, complex numbers, standard and polar form, complex plane, algebra of complex numbers, complex conjugate

2. Functions: general definition; modulus function; polynomials; drawing real, complex, and parametric functions; combining functions; inverse functions; trigonometric functions; function properties: continuity, symmetry, periodicity, and monotonicity

3. Limits: fundamental notion of a limit and definition of a function limit; infinite; discontinuity; left and right limitations; limits of functions in combination

4. Power series: the limit of a sum of polynomials; convergence; exponential functions described as power series; power series formulations for trigonometric functions; hyperbolic functions and their connection to trigonometric functions

5. More Complex Numbers: Euler’s formula and de Moivre’s theorem; exponential form of complex numbers; roots of unity; algebraic basic theorem; complex number transformations

6. Differentiation: basic concept of derivative; l’Hopital’s rule; sums, products, quotients, and chain rule; inverse function derivatives; turning and critical points; Taylor series; radius of convergence; implicit and parametric function derivatives; logarithmic differentiation

7. Integration: definite and indefinite integrals; calculus’ basic theorem; proper and improper integrals; integration methods; curve lengths, surfaces, and revolution volumes.

8. Two- and three-dimensional vectors: Cartesian coordinates, position vectors; vectors as directed line segments (magnitude, direction); addition, subtraction; scalar multiplication; unit vectors; inner/dot product; angle between two vectors; orthogonality; components and projections; extension to higher dimensions; vector/cross product in three dimensions; scalar triple product and volume; vector representation of points and lines Orthogonal coordinate systems are a kind of coordinate system in which the coordinates (polar, parabolic, elliptic, cylindrical and spherical).

9. Multivariable functions: partial derivative (definition, evaluation); chain rule, higher partial derivatives, including mixed derivatives; higher dimensional Taylor expansion; gradient vector (grad); critical or turning points (maxima, minima, saddle-points); relationship with contour lines and gad; identifying critical points; derivative test and use of the discriminant; tu

10. Multiple Integrals: area, scalar-valued function integration in the plane; order of integration; area integral in polar coordinates; Jacobian and change of variable; directed path or line integral of a vector-valued function in the plane; path-dependence/independence; conservative vector fields; potential function

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