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(All Edits done in bold)

Student years[ edit]
In April 1869, following Sofia's and Vladimir's brief stay in Vienna, where she attended lectures in physics at the university, they moved to Heidelberg. Through great efforts, she obtained permission to audit classes with the professors' approval at the University of Heidelberg. There she attended courses in physics and mathematics under such teachers as Hermann von Helmholtz, Gustav Kirchhoff and Robert Bunsen. Vladimir, meanwhile, went on to the University of Jena to pursue a doctorate in paleontology.

In October 1869, shortly after attending courses in Heidelberg, she visited London with Vladimir, who spent time with his colleagues Thomas Huxley and Charles Darwin, while she was invited to attend George Eliot's Sunday salons. There, at age nineteen, she met Herbert Spencer and was led into a debate, at Eliot's instigation, on "woman's capacity for abstract thought". Although there is no record of the details of their conversation, she had just completed a lecture course in Heidelberg on mechanics, and she may just possibly have made mention of the Euler equations governing the motion of a rigid body (see following section). George Eliot was writing Middlemarch at the time, in which one finds the remarkable sentence: "In short, woman was a problem which, since Mr. Brooke's mind felt blank before it, could hardly be less complicated than the revolutions of an irregular solid." This was well before she made her notable contribution of the "Kovalevskaya top" to the brief list of known examples of integrable rigid body motion (see following section).

In October 1870, Kovalevskaya moved to Berlin, where she began to take private lessons with Karl Weierstrass, since the university would not allow her even to audit classes. He was very impressed with her mathematical skills, and over the subsequent three years taught her the same material that comprised his lectures at the university.

In 1871 she briefly traveled to Paris together with Vladimir in order to help in the Paris Commune, where Kovalevskaya attended the injured and her sister Anyuta was active in the Commune. With the fall of the Commune, however, both Anyuta and her common law husband Victor Jaclard, who was leader of the Montmartre contingent of the National Guard and a prominent Blanquiste, were arrested. Although Anyuta managed to escape to London, Jaclard was sentenced to execution. However, with the assistance of Sofia's and Anyuta's father General Krukovsky, who had come urgently to Paris to help Anyuta and who wrote to Adolphe Thiers asking for clemency, they managed to save Victor Jaclard.

Kovalevskaya returned to Berlin and continued her studies with Weierstrass for three more years. In 1874 she presented three papers—on partial differential equations, on the dynamics of Saturn's rings, and on elliptic integrals—to the University of Göttingen as her doctoral dissertation. With the support of Weierstrass, this earned her a doctorate in mathematics summa cum laude, after Weierstrass succeeded in having her exempted from the usual oral examinations.

Last years in Germany and Sweden[ edit]
Bust by Finnish sculptor Walter Runeberg Kovalevskaya's grave, Norra begravningsplatsen

In 1874, Kovalevskaya and her husband Vladimir returned to Russia, but Vladimir failed to secure a professorship because of his radical beliefs. (Kovalevskaya never would have been considered for such a position because of her gender.) During this time they tried a variety of schemes to support themselves, including real estate development and involvement with an oil company. But in the late 1870s they developed financial problems, leading to bankruptcy.

In 1875, for some unknown reason, perhaps the death of her father, Sofia and Vladimir decided to spend several years together as an actual married couple. Three years later their daughter, Sofia (called "Fufa"), was born. After almost two years devoted to raising her daughter, Kovalevskaya put Fufa under the care of relatives and friends, resumed her work in mathematics, and left Vladimir for what would be the last time.

Vladimir, who had always suffered severe mood swings, became more unstable. In 1883, faced with worsening mood swings and the possibility of being prosecuted for his role in a stock swindle, Vladimir committed suicide.

That year, with the help of the mathematician Gösta Mittag-Leffler, whom she had known as a fellow student of Weierstrass, Kovalevskaya was able to secure a position as a privat-docent at Stockholm University in Sweden. Kovalevskaya met Mittag-Leffler's sister, the actress, novelist, and playwright Anne Charlotte Edgren-Leffler. Until Kovalevskaya's death the two women shared a close friendship.

In 1884 Kovalevskaya was appointed to a five-year position as Extraordinary Professor (assistant professor in modern terminology) and became an editor of Acta Mathematica. In 1888 she won the Prix Bordin of the French Academy of Science, for her work "Mémoire sur un cas particulier du problème de la rotation d'un corps pesant autour d'un point fixe, où l'intégration s'effectue à l'aide des fonctions ultraelliptiques du temps". Her submission featured the celebrated discovery of what is now known as the "Kovalevskaya top", which was subsequently shown to be the only other case of rigid body motion that is "completely integrable" other than the tops of Euler and Lagrange.

In 1889 Kovalevskaya was appointed Ordinary Professor (full professor) at Stockholm University, the first woman in Europe in modern times to hold such a position. After much lobbying on her behalf (and a change in the Academy's rules) she was made a Corresponding Member of the Russian Academy of Sciences, but she was never offered a professorship in Russia.

Kovalevskaya, who was involved in the vibrant, politically progressive and feminist currents of late nineteenth-century Russian nihilism, wrote several non-mathematical works as well, including a memoir, A Russian Childhood, two plays (in collaboration with Duchess Anne Charlotte Edgren-Leffler) and a partly autobiographical novel, Nihilist Girl (1890).

In 1889, Kovalevskaya fell in love with Maxim Kovalevsky, a distant relation of her deceased husband, but insisted on not marrying him because she would not be able to settle down and live with him.

Kovalevskaya died of influenza complicated by pneumonia in 1891 at age forty-one, after returning from a vacation in Nice with Maxim. She is buried in Solna, Sweden, at Norra begravningsplatsen.

Contributions to mathematics[ edit]
'''Sofya Kovalevskaya developed many different mathematic equations and theorems during the course of her career. One of the theorems accredited to her name is the Cauchy-Kovalevskaya theorem. This theory is part of partial differential equations and is used in branches of math such as calculus. Kovalevskaya also contributed to the theorem of power series; she specifically looked at power series with two variables. Later in life Kovalevskaya would use the Cauchy-Kovalevskaya theorem to give a new derivation of Bruns's Theorem. Before passing away Kovalevskaya developed a more realistic model of Lame's differential equation for the propagation of a wave in a solid medium.  '''