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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.

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.

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.

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.

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.

Vertebrates are amongst the most successful animal groups. From fish, amphibians, lizards, crocodiles, birds and mammals, you will gain an understanding how the basal members of the clade have diversified and evolved to fill every imaginable niche on land, sea and air.
You will develop an overview of the anatomy, physiology, behaviour and the ecological interactions across this group of vertebrates.

Vertebrates are amongst the most successful animal groups. From fish, amphibians, lizards, crocodiles, birds and mammals, you will gain an understanding how the basal members of the clade have diversified and evolved to fill every imaginable niche on land, sea and air.
You will develop an overview of the anatomy, physiology, behaviour and the ecological interactions across this group of vertebrates.