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This module takes an empirical approach to questions such as:

- Are there patterns of speech and language associated with males and females in varieties of English?
- What is the role of teenagers in the propagation of change in English?
- After a certain age are our accents ‘set’ or can we change over our lifespans?
- How do migration and language contact lead to the birth of new English dialects?

The module provides a comprehensive introduction to the sociolinguistic paradigm – the quantitative approach to linguistic variation. Through reference to seminal studies, as well as recent advances in the field, we examine how social factors such as age, gender, ethnicity and social network etc. impact on patterns of variation and change in English.

This module takes an empirical approach to questions such as:

- Are there patterns of speech and language associated with males and females in varieties of English?
- What is the role of teenagers in the propagation of change in English?
- After a certain age are our accents ‘set’ or can we change over our lifespans?
- How do migration and language contact lead to the birth of new English dialects?

The module provides a comprehensive introduction to the sociolinguistic paradigm – the quantitative approach to linguistic variation. Through reference to seminal studies, as well as recent advances in the field, we examine how social factors such as age, gender, ethnicity and social network etc. impact on patterns of variation and change in English.

In the first part of this module we build on multivariate calculus studied in the first year and extend it to the calculus of scalar and vector functions of several variables. Line, surface and volume integrals are considered and a number of theorems involving these integrals (named after Gauss, Stokes and Green) will be discussed. In particular Green’s theorem, which gives a formula for the line integral of a vector field in the plane round a closed curve, is closely related to complex integration considered in the second part of the module. The integral theorems are also useful in many branches of Applied Mathematics and to describe physical quantities that vary in space and in time. For example, this module is a pre-requisite for MATH2044, Fields and fluids, where these methods are used to describe the behaviour of fluids and of electromagnetic fields.

In the second part of this module, we extend our investigation of calculus to functions of a complex variable, once again building on the material studied in the first year. This theory has both great aesthetic appeal and a large number of applications. We focus here on the integration of these functions, particularly along curves in the complex plane. We develop the basic theory and ideas of the integration of a function of a complex variable, use the main theorems such as Cauchy’s theorem and the Cauchy integral formula, and explore some of their consequences, such as the Fundamental Theorem of Algebra and the evaluation of real integrals.

In the first part of this module we build on multivariate calculus studied in the first year and extend it to the calculus of scalar and vector functions of several variables. Line, surface and volume integrals are considered and a number of theorems involving these integrals (named after Gauss, Stokes and Green) will be discussed. In particular Green’s theorem, which gives a formula for the line integral of a vector field in the plane round a closed curve, is closely related to complex integration considered in the second part of the module. The integral theorems are also useful in many branches of Applied Mathematics and to describe physical quantities that vary in space and in time. For example, this module is a pre-requisite for MATH2044, Fields and fluids, where these methods are used to describe the behaviour of fluids and of electromagnetic fields.

In the second part of this module, we extend our investigation of calculus to functions of a complex variable, once again building on the material studied in the first year. This theory has both great aesthetic appeal and a large number of applications. We focus here on the integration of these functions, particularly along curves in the complex plane. We develop the basic theory and ideas of the integration of a function of a complex variable, use the main theorems such as Cauchy’s theorem and the Cauchy integral formula, and explore some of their consequences, such as the Fundamental Theorem of Algebra and the evaluation of real integrals.

In the first part of this module we build on multivariate calculus studied in the first year and extend it to the calculus of scalar and vector functions of several variables. Line, surface and volume integrals are considered and a number of theorems involving these integrals (named after Gauss, Stokes and Green) will be discussed. In particular Green’s theorem, which gives a formula for the line integral of a vector field in the plane round a closed curve, is closely related to complex integration considered in the second part of the module. The integral theorems are also useful in many branches of Applied Mathematics and to describe physical quantities that vary in space and in time. For example, this module is a pre-requisite for MATH2044, Fields and fluids, where these methods are used to describe the behaviour of fluids and of electromagnetic fields.

In the second part of this module, we extend our investigation of calculus to functions of a complex variable, once again building on the material studied in the first year. This theory has both great aesthetic appeal and a large number of applications. We focus here on the integration of these functions, particularly along curves in the complex plane. We develop the basic theory and ideas of the integration of a function of a complex variable, use the main theorems such as Cauchy’s theorem and the Cauchy integral formula, and explore some of their consequences, such as the Fundamental Theorem of Algebra and the evaluation of real integrals.

In the first part of this module we build on multivariable calculus studied in the first year and extend it to the calculus of scalar and vector fields. Cartesian as well as curvilinear coordinates are used, and we study gradient, divergence and curl. Line, surface and volume integrals over scalar and vector fields are studied in detail. The most important results are the integral theorems by Stokes and Gauss, which bring together nearly all concepts studied in the first part of the module. As a corollary, Green’s theorem is derived, which is closely related to complex integration considered in the second part of the module. The integral theorems are essential in many branches of Applied Mathematics. For example, this module is a pre-requisite for the module Fields and Fluids, where the techniques learned here are employed to describe the behaviour of fluids and of electromagnetic fields.

In the second part of this module, we extend our investigation of calculus to functions of a complex variable, once again building on the material studied in the first year. This theory has both great aesthetic appeal and a large number of applications. We study differentiability of complex functions and then focus on integration along curves in the complex plane, discussing Cauchy's theorem and integral formula. Series expansion of complex functions is developed and then used to classify singularities and define the residue. This leads to the residue theorem, which is employed in many examples, in particular for the evaluation of real integrals. Complex variable theory is crucial for various applications in Applied Mathematics, in particular Theoretical Physics, and elements of it will be used in Fields and Fluids.

In the first part of this module we build on multivariable calculus studied in the first year and extend it to the calculus of scalar and vector fields. Cartesian as well as curvilinear coordinates are used, and we study gradient, divergence and curl. Line, surface and volume integrals over scalar and vector fields are studied in detail. The most important results are the integral theorems by Stokes and Gauss, which bring together nearly all concepts studied in the first part of the module. As a corollary, Green’s theorem is derived, which is closely related to complex integration considered in the second part of the module. The integral theorems are essential in many branches of Applied Mathematics. For example, this module is a pre-requisite for the module Fields and Fluids, where the techniques learned here are employed to describe the behaviour of fluids and of electromagnetic fields.

In the second part of this module, we extend our investigation of calculus to functions of a complex variable, once again building on the material studied in the first year. This theory has both great aesthetic appeal and a large number of applications. We study differentiability of complex functions and then focus on integration along curves in the complex plane, discussing Cauchy's theorem and integral formula. Series expansion of complex functions is developed and then used to classify singularities and define the residue. This leads to the residue theorem, which is employed in many examples, in particular for the evaluation of real integrals. Complex variable theory is crucial for various applications in Applied Mathematics, in particular Theoretical Physics, and elements of it will be used in Fields and Fluids.

In the first part of this module we build on multivariable calculus studied in the first year and extend it to the calculus of scalar and vector fields. Cartesian as well as curvilinear coordinates are used, and we study gradient, divergence and curl. Line, surface and volume integrals over scalar and vector fields are studied in detail. The most important results are the integral theorems by Stokes and Gauss, which bring together nearly all concepts studied in the first part of the module. As a corollary, Green’s theorem is derived, which is closely related to complex integration considered in the second part of the module. The integral theorems are essential in many branches of Applied Mathematics. For example, this module is a pre-requisite for the module Fields and Fluids, where the techniques learned here are employed to describe the behaviour of fluids and of electromagnetic fields.

In the second part of this module, we extend our investigation of calculus to functions of a complex variable, once again building on the material studied in the first year. This theory has both great aesthetic appeal and a large number of applications. We study differentiability of complex functions and then focus on integration along curves in the complex plane, discussing Cauchy's theorem and integral formula. Series expansion of complex functions is developed and then used to classify singularities and define the residue. This leads to the residue theorem, which is employed in many examples, in particular for the evaluation of real integrals. Complex variable theory is crucial for various applications in Applied Mathematics, in particular Theoretical Physics, and elements of it will be used in Fields and Fluids.

In the first part of this module we build on multivariable calculus studied in the first year and extend it to the calculus of scalar and vector fields. Cartesian as well as curvilinear coordinates are used, and we study gradient, divergence and curl. Line, surface and volume integrals over scalar and vector fields are studied in detail. The most important results are the integral theorems by Stokes and Gauss, which bring together nearly all concepts studied in the first part of the module. As a corollary, Green’s theorem is derived, which is closely related to complex integration considered in the second part of the module. The integral theorems are essential in many branches of Applied Mathematics. For example, this module is a pre-requisite for the module Fields and Fluids, where the techniques learned here are employed to describe the behaviour of fluids and of electromagnetic fields.

In the second part of this module, we extend our investigation of calculus to functions of a complex variable, once again building on the material studied in the first year. This theory has both great aesthetic appeal and a large number of applications. We study differentiability of complex functions and then focus on integration along curves in the complex plane, discussing Cauchy's theorem and integral formula. Series expansion of complex functions is developed and then used to classify singularities and define the residue. This leads to the residue theorem, which is employed in many examples, in particular for the evaluation of real integrals. Complex variable theory is crucial for various applications in Applied Mathematics, in particular Theoretical Physics, and elements of it will be used in Fields and Fluids.

This module provides the second year student with the basic concepts of human and other vertebrate animal development. Students will come to understand the main mechanisms behind both animal development and organised cellular differentiation and how these processes are studied. They will also become aware of how various changes in developmental pathways can play a role in human and animal health.

Lectures will be accompanied by practicals, some of which involve the use of animal tissue, with alternatives in place if required to meet minimum learning outcomes.